No, you've not laid out the logic by which that statement follows from the others.
> America with its democracy exporting foreign policy gives the rational that democracies don't go to war.
I'd actually argue quite a lot with this: the US does not export democracy, it exports capitalism and calls it democracy. It's quite happy to overthrow democratic socialist governments.
Really the key factor is that trading partners don't go to war, because then it's unprofitable and/or they don't have enough independent infrastructure to survive. You can't go to war with your nose to spite your face. This is partly why, despite the bad relations between the EU and Russia and the small war in Ukraine, the gas pipeline remains untouched. They need the money and we need the gas.
And your inference that I'm American is also wrong. I'm an Englishman living in Scotland supporting Scottish independence within the EU. In fact, I spent a lot of time trying to be a "reasonable" Euroskeptic for years, but this year of the referendum I gave up because the unreasonable side has taken over. Yes, the EU is not very democratic and parts of the EU get economically steamrollered. Guess what - this happens within countries too. Greece is to the EU as 80s Liverpool and the "north" was to the Thatcher government: abandoned.
Leaving the EU does not make Europe go away, it doesn't solve any of the economic or social problems, but it does remove a mechanism for resolving them peacefully.
The UK is not very democratic on close inspection either, and all these threats to remove one of the few bits of black-letter constitutional law we do have - ECHR - are really not encouraging.