On your second paragraph, once we have 'taken back control' (to use a cliché), international trade policy will once again fall within the competence of our Westminster parliament.
Assuming the government of the time doesn't decide it falls under royal prerogative.
In the EU tarrifs are subject to the
1) democratically elected EU parliament (where the makeup actually reflects the population, rather than having a party representing the views of 34% of those that voted having all the power)
2) a commission (the executive), appointed by each member government, in the same way that ministers are appointed, and no-less democratic than having Boris Johnson as foreign secretary.
3) the council of the EU, which comprises the head of government of each country (in our case May)
People in the UK can vote for the EU parliament based, in part, on their international trade policy, they can also vote for their own country's government based, in part, on their international trade policy.
So your point seems to come down to it's ok to have a common policy for disparate economies such as the banking sector in london, the fishing sector in Cornwall, and the boat building sector in Glasgow, but it's not ok to incorporate the rest of europe into those policies - despite the similarities the fishing sector in Cornwall has to the fishing sector on the other side of the channel.
I refuse to believe your argument is that we don't have a democratic say in the EU, just that what's good for the EU as a whole isn't necessarily good for the UK. That argument of course works in the UK too -- what's good for the UK as a whole isn't necessarily good for Wales. That argument of course works in Wales too -- what's good for Wales as a whole isn't necessarily good for Bangor.
This was exemplified in March when Wales was struggling with Chinese steel being dumped on the world market at tiny rates. We decided as a country that it would be better for the UK (but worse for parts of Wales) to allow this dumping to continue. Democracy.
As it happens the rest of the EU was also struggling with this dumping, and the EU attempted to bring in tarrifs, but failed because of Osborne's "love affair with China" meaning that Cameron managed to block the tariffs (see
http://www.telegraph.co.uk/business...es-steel-industry-to-curry-favour-with-china/ for the story)
In a post-UK EU, the EU would have brought these tarrifs in, and the UK would have not had the tarriffs. This would mean the UK could buy steel cheaply from China. But then what? We could sell that steel to europe, working around the EU tarrifs, and taking a cut. That would be great, but I don't think the EU would be happy with that, so would presumably also put tarrifs on UK steel. OK, so we don't re-export it. Instead we use it to build cars cheaper than France can (as the steel would be cheaper). How would the EU respond to our undercutting there? They'd have to put a tarrif on our car exports that use steel from china and are exported into the EU. Perhaps we retaliate with tarrifs on crappy Italian wine, and then the EU retaliates with tarrifs on our food exports.
Oh what a tangled web we weave.