And if you try to hold a referendum on joining NATO, Russia will be all over it like nobody's business. I'm glad that we decided to not have a referendum and considering how pro-NATO the opinion polling is, I'm not really worried for the death of democracy.
Referenda have always been double-edged swords. They’ve been using by totalitarian populists since at least the Roman republic.
In most representative democracies, treaties have to be approved after the fact but they certainly do not need to be pre-approved before being negotiated. This is not more undemocratic than a spur-of-the-moment referendum.
If you don't move quick, you won't move. This is the purpose of representative democracy - to allow trusted leaders to make decisions like this on your behalf.