I stopped reading once you made a distinction between “tariffs” and “politics”: you don’t have one without the other. You have a common market only if you can agree common rules, and you agree common rules with shared political institutions.
You cannot buy chlorine-washed chicken in the EU because EU countries agreed that, in order to be able to freely trade chicken in the union, chicken should have certain qualities - qualities defined through a political process.
The EU is simply a place where all the stuff that used to get done with slow and opaque bilateral treaties or with wars, now gets done mostly in the open, faster and more democratically than before. It doesn’t “hold free trade hostage”, it’s actually the opposite: it takes free trade to its eventual consequences. Everything is free to trade, all the time; but to do that, we must agree on common standards to make, sell, buy, and consume all stuff - it wouldn’t be fair competition, otherwise, right?
So we built a process to write these rules together. Once we have rules, we need ways to enforce them and amend them - which happen to be useful to also sort out other pesky issues like borders and to foster collaboration in various endeavours. And if all products and services can be freely moved across the market, why shouldn’t people be allowed to do the same?
That’s the EU, in a nutshell; built on and for commerce. Trying to unpick trade from the rule-writing is nonsensical.