David Lidington (sometimes described as Theresa May’s Deputy PM) was Minister for Europe in the coalition government between 2010 and 2015. Later he was promoted to Justice Secretary and Lord Chancellor. He remains firmly pro-remain and well-established in the UK Government. Indeed he gave his support in the process of replacing Theresa May to Rory Stewart, the only candidate who actively supports the case for the UK to remain in the EU.
Lidington gave a speech to the UK Association for European Law on 25 November 2010 which, given his seniority in the current Conservative Government, highlights some important issues:
“…in the 38 years since the European Communities Act  became law, we have seen the influence of European law upon our own jurisprudence become greater and more pervasive.”
“That has been the logical consequence of incorporating the doctrines of direct effect and the primacy of European law into our own system. … Lord Denning was clear that European law was binding because a sovereign Parliament had decided that it should be so.” 
“The principle of parliamentary sovereignty, the doctrine that European law has effect here for one reason only, namely that that authority has been conferred upon it by Acts of Parliament and that its authority subsists only for as long as Parliament so decides, has been upheld consistently by our courts.”
UK courts have no choice, that much is clear; the responsibility lies, as he says, with the UK Parliament.
“But what is also true is that, in recent years, the argument has been advanced that we are now in a different legal order and that European law has now acquired an autonomous authority which has superseded the traditional doctrine of parliamentary sovereignty.”
This argument was advanced most vigorously by the prosecution in the so-called Metric Martyrs case (Thoburn v Sunderland City Council) in 2002. On that occasion, the court rejected the plea, instead asserting, in the words of Lord Justice Laws, that,
“Parliament cannot bind its successors by stipulating against repeal, wholly or partly, of the European Communities Act. It cannot stipulate as to the manner and form of any subsequent legislation. It cannot stipulate against implied repeal any more than it can stipulate against express repeal. Thus there is nothing in the European Communities Act which allows the Court of Justice or any other institutions of the EU, to touch or qualify the conditions of Parliament’s legislative supremacy in the United Kingdom.”
“Let me be clear that this Government believes that it is in the national interest of the United Kingdom to be a member of the European Union and to be a very active and positive participant in its work.”
No surprise there, and the view of the government hasn’t changed since 2010.
“But if we are going to make that case successfully, we need to do more to address the deep felt sense of disconnection which exists between the British people and the institutions and decision-making processes of the EU.”
He recognised the problem well before the 2016 referendum; his ‘solution’ is outlined below, with some sceptical comments from us.
“The European Union Bill…sets out quite deliberately to give new powers both to the British people and to Parliament to hold this and future governments to account for how Minsters take decisions on behalf of the United Kingdom within Europe.”
“The Bill will require that if in the future there is a proposal to amend the EU treaties or to propose a new replacement treaty which would constitute a transfer of power or competence from the UK to the EU, the government of the day would have to have the consent of the British people in a national referendum before that could be agreed and that treaty or treaty amendment ratified. There will be a referendum lock on any such treaty or treaty amendment, a lock to which only the British people will hold the key.”
The European Union Act 2011 was proposed by the coalition government in 2010 and passed by Parliament in March 2011 as a reaction to the European Union (Amendment) Act 2008, which had instituted the Treaty of Lisbon without participation by the Labour Prime Minister of the day, Gordon Brown, and with no referendum although one had been promised in 2005 in the Labour manifesto. The only referendum that the British people have been offered, despite this legal requirement and a deluge of laws and regulations continuing pour down from the EU, was the 2016 chance to leave the EU. After the Referendum the 2011 Act was repealed and replaced by the European Union (Withdrawal) Act 2018.
“That habit of participatory democracy…generates an expectation that this is the right way in which collective decisions should be taken. Conversely, a sense that the really important decisions about the way in which this country is governed are taken by politicians, perhaps with the use of heavy parliamentary whipping and curtailed debate through the use of the guillotine, reinforces public alienation from the political process. There is no doubt in my mind that one reason for the depth of public disaffection from Europe in this country is the powerful sense of resentment felt by British people at having been denied a referendum on the Lisbon Treaty although our nearest neighbours in the Irish Republic were able to have their say.” 
Next come some efforts to persuade us that the UK government has full control over its democratic and legal processes, which is contradicted by the reality of EU membership (and which he wishes to maintain).
“So one change we are making in the European Union Bill is to ensure that in future any amendment to the Treaty on the European Union or to the Treaty on the Functioning of European Union, under either revision procedure, would require primary legislation, a full Act of Parliament, before the United Kingdom would be able to ratify that change.”
“Since Lisbon, the different types of EU competence and the extent of each type of competence has been set out clearly in the Treaties and any extension of competence would under our new legislation trigger a referendum.” 
“The Bill places an absolute and unqualified referendum lock on the transfer of competence, the creation of new EU competence, or the removal of limits to existing competences and also upon a whole raft of specified policy areas.” 
“The EU Bill will require Parliament to approve each use of a ratchet clause* in an Act of Parliament before the Government can agree to their use in the Council or the European Council; an increased level of parliamentary control which will allow both Houses the opportunity to consider each measure carefully.”
“And, as with the treaty changes, if any such measure based on one of the passerelle clauses were to propose a transfer of power or competence, the Bill makes clear that a referendum of the British people will be required.” (our emphasis)
“These provisions provide Parliament and the British people with a key role in determining whether the future use of any of these so-called self-amending provisions in the Treaties can be used; and is a clear demonstration of the Government’s commitment to rebuild trust and reconnect the British public with decisions taken in Brussels.”
Clearly a majority of British voters didn’t believe him or his Government. If either the UK or the EU were becoming more prosperous, and our democracy were restored, perhaps in the way he suggests, then maybe there would be a good argument for remaining a member; neither applies in practice.
*“The Bill will also give Parliament more control over whether the Government can agree to a number of other important EU decisions, sometimes referred to as the self-amending provisions of the Lisbon Treaty. These decisions, known as ‘passerelles’ or ‘ratchet clauses’, contain built-in mechanisms, which allow for modifications to the EU Treaties or the exercise of one-way options without recourse to either of the formal methods of Treaty change…”.
 The Withdrawal Method-2 (25/06/2019)
 Themes-15: Democracy (10/06/2019)
 Subsidiarity and Competence (29/08/2018)
 More is Less: Subsidiarity (1 & 2) (26/10/2018)
There is more:
“This is, I believe, an ambitious and important piece of legislation. It will give powers to the British people and to Parliament which they have never previously enjoyed in respect of decisions about our engagement with the European Union. The Bill will also give people the assurance they are entitled to expect that the sovereignty of Parliament and the ultimate right of the people themselves to decide which powers should be delegated for collective decisions within Europe and which should be maintained as distinct national competences, are both being properly safeguarded. And given those safeguards, I believe that people in the United Kingdom will be more willing to support policies and positive work within the EU to defend and promote the interests of our country.”
“It is in our interests as a country for the nations of Europe to work together to enhance the prosperity and security of our continent and for us in Britain to be an active player in shaping the priorities of the European Union. But it is for the British people to determine the nature and extent of the powers that they wish to share with our European neighbours. What our new legislation seeks to do is to ensure that it is the people ultimately who retain the power to decide.”
The British people may have been given more powers by the 2011 Act but governments of that day and since have not chosen to exercise those powers on behalf of ‘their’ people. It would be in the UK’s interest to enhance our prosperity and security (though he says “our continent“). Lidington’s references to “safeguards” did not inspire the people to support the EU as he wished, perhaps because his government did not, in practice, respect and act on those supposed safeguards.