As part of leaving you'd be cut off from the ECB and that connection is required for the banking system to operate.

But Montenegro's banking system works. They aren't part of the ECB.

They never formally joined the Eurozone to begin with, they just allow the Euro to circulate and accept it for payment. It's not even legal tender there, in theory. The integration issues are more to do with how the banking systems of formally Eurozone integrated countries are wired up to the ECB and how countries come to rely on it for "liquidity" (money printing to pay for expensive large government).

Also note that the EU don't like Montenegro using the Euro and tolerate it only because it's tiny/doesn't matter.

A leaving country could keep using the Euro while working on the transition back to their own currency. When the USSR collapsed - its former members continued to use Soviet rubles for some time until they transitioned to their own currencies.

So yeah - if a country really wanted to leave - a shared currency would not stop it.

It depends on how much cooperation the EU would extend (assume zero). For example paying other companies in euros requires transactions with the central bank. If the EU don't want you to leave they can just say, well, the moment you declare you're out we cut you off from the banking system and you can't settle your government debts anymore because you can't make euro transactions. For example.

Yeah, but I don't think that they would do it. We already had Brexit and the UK was allowed to leave. Had they not colonised Northern Ireland - the leaving process would have been even easier.

UK has its own currency and independent central bank, also it was a net payer into the EU budget so there was limited leverage.