The same economic theories that lead to the conclusion that low tariffs are good place the same importance on the free exchange of goods as the free exchange of labour. The treaty of Rome is the governing treaty in the EU in this respect and it has provisions on both. Since it is precisely the free exchange of labour that the UK wants to eliminate, they will need to violate the treaty and lose the free exchange of goods. How this will affect the application in the UK of existing trade agreements between external countries and the EU is another question. Since the bureaucratic process of bringing goods to the UK will be different than bringing it to the EU, foreign countries will likely take the stance that those trade agreements no longer apply to the UK.
That's not true. It's only true if both sides can compete on equal terms. The EU guarantees that. Without Britain in the EU, there will be situations where tariffs are beneficial for one side.
"Generally" strongly assumes equal or even perfect competition.
For example say Country A and B trade coal and agree to no tariffs. Country A now cuts back on worker protection and wages, such that their coal is cheaper. Country B's only other chance than doing the same would be to raise a tariff to keep its own coal competitive. That's why Country B wants a say in Country A's labor laws.
And this is exactly the reason for a supra-national body that comes in with annoying regulations and red tape that stops you from competing with Johnny Foreigner in the full-blooded way you want to.
You're confusing economic theory with reality. There were already plenty of conflicts in many sectors (eg fishing) and now each EU country that think they can gain something in any sensitive sector by having tariffs on UK products will push in that direction.