Loss of sovereignty in particular was a fictional concern in regards to the EU, given the structure of the EU and the relatively high power of the UK within it. The degree to which the EU should be made more democratic is precisely the degree to which it remains exactly what leave campaigners said they wanted to replace it with: a traditional boring free trade agreement in the hands of negotiators appointed by the governments of the member states.
"High" migration likewise had nothing much to do with EU membership, as the government demonstrated precisely by following Brexit with, as you say, record high immigration.
One of the other famous big concerns Leave campaigners had was the cost, which famously became the £350 million a week on the side of a bus. This number was even called out as a falsehood at the time, but it was believed by enough to make a difference.
Remainers were unable to convince the majority that the benefits of EU membership was worth the cost, financial or otherwise, regardless.