Don't take my word for its being a myth, here's a column from that bastion of Remain, The Telegraph:
No, Britain wasn't lied to when we joined the EU. We knew what we were getting into
Nowhere has grievance festered more than over the contention from some Brexiteers that implications of EU membership for British sovereignty were deliberately concealed from voters by deceitful 1970s politicians.
Columnist Melanie Philips has asserted that Edward Heath "knew full well that if the British public understood the implications for national self-government, they would never agree to membership of the European club.
Echoing her, commentator William Cash, son of Conservative MP Sir William Cash, attacked the treachery and political deceit of what really went on in the political backrooms on London and Brussels...it is clear that our senior politicians knew more than they were letting on about the sovereignty implications when they negotiated to join the EEC."
But a cursory read of the actual parliamentary debates tells a different story
https://www.telegraph.co.uk/opinion/2016/03/29/no-britain-wasnt-lied-to-when-we-joined-the-eu-we-knew-what-we-w/
If you read through it, it shows that the question of pooled versus national sovereignty was prominent in parliamentary debates in 1971, so it was always framed as more than a "trading agreement".
Here's a summary of a Chatham House paper that discussed UK sovereignty in the EU in the run-up to the 2016 referendum:
Britain, the EU and the Sovereignty Myth
The question of sovereignty lies at the heart of the UKs upcoming EU referendum. Many in Britain believe that the process of EU decision-making has undermined British parliamentary democracy, and that leaving the EU is the only way for the British people to regain control of their sovereignty.
This ignores the fact that successive British governments have chosen to pool aspects of the countrys sovereign power in the EU in order to achieve national objectives that they could not have achieved on their own, such as creating the single market, enlarging the EU, constraining Irans nuclear programme, and helping to design an ambitious EU climate change strategy.
Apart from EU immigration, the British government still determines the vast majority of policy over every issue of greatest concern to British voters including health, education, pensions, welfare, monetary policy, defence and border security. The arguments for leaving also ignore the fact that the UK controls more than 98 per cent of its public expenditure.
...
In a world that is more interdependent today than it was when the UK joined the European Economic Community in 1973, the notion of absolute British sovereignty is illusory. It is also worthless if it limits the ability of future British governments to ensure the security and prosperity of their citizens. Judging from the UKs experience and its future prospects, the opportunities from remaining in the EU far outweigh the risks of doing so, and the risks of leaving far outweigh the opportunities.
https://www.chathamhouse.org/publication/britain-eu-and-sovereignty-myth
The summary touches on the issue of EU immigration and the social pressures it can cause, but doesn't expand on the fact that successive governments have chosen not to impose allowable limitations on EU immigration, partly because it would involve relatively costly bureaucracy, but not least because such immigration is a net benefit to the country, as is our reciprocal freedom of movement.
I'll leave Muriel to decide whether to pick up on your other points since you haven't bothered to respond to my suggestion below that you research just how much influence the UK has had on EU legislation over the years.