Re: The elephant in the room - stagnant wages because of the free movement of labour
Maybe I'm just too ignorant of economics: Simple logic tells me that if countries wanted to sell stuff more cheaply they would already be doing so. And if they wanted to pay more....
FWIW on this, the EU is a customs unions, external tariffs are set by the EU and if the EU takes a disliking to a country the UK does a lot of trade with (or wants to do a lot of trade with) or fails at negotiating low tariff or tariff-free deals with that the UK trades heavily with then the UK gets screwed.
China is your classic case of both, UK was happily buying solar panels from China trying to get some low carbon infrastructure going. Some German (and I think Spanish) trade groups got all hot under the collar, protested to both the German govt/EU and the EU slapped a huge tariff on Chinese solar panels without much investigation because the Germans didn't feel like being competitive. Ironically the German govt later said they regretted the whole thing because it turned out the Germans were doing the same as the UK and buying from China rather than German suppliers and nothing they could do would get rid of it. Which is circular to another problem with the EU massive inertia/momentum depending on the situation - that's not surprising because the EU is huge and it's member states are very different but that's a problem they've made no attempt to resolve, it's not even clear they recognise it is one. The EU isn't seriously looking at a deal with China and when they do it could easily take 10 years to sort, again because of the sizes of both sides.
The US is another example of UK trade harmed by the EU. The UK wants a trade deal with the US; it has done for decades. The UK pushed the EU-US deal and it got turned into the 7-headed monster from the deep now known as TTIP. It's not surprising because both sides of the deal are massive - but it's so big and so comprehensive (excessively so) that even before the brexit vote it looked somewhat dead in the water and post that and post-Obama I can't see it surviving. Talk about all your eggs in one basket. It's hard to even tell which side of the Atlantic hates it more so perhaps it should be put out of its misery.
There's loads of examples of this all over the EU, there was the one where T&L can't import cane sugar from poor states because of a combination of subsidies on sugar beet which we *really* shouldn't be subsidising for a long list of reasons including but not limited to - it's sugar; and the fact that the EU slaps tariffs on those imports because no logical reason; which is not only damaging to the EU via budgets and via health (sugar is far too cheap in Europe which is why you hear about countries looking at adding tax to it all the time which is completely illogical) - but is also damaging to poorer countries that could be doing things like growing sugar to export to richer countries to help make them less dependant on aid.
I could go on all day with examples that even I know about and I assume what I know isn't even the half of it.