Purely speaking as a Czech and European Union citizen now, we literally didn't have to join either, and if the populace were to become against it overall, we can at any time have a referendum to leave either, Brexit style. The reason for that is that neither Brussels, nor Washington would send tanks here if we did so.
There would be some economic drawbacks to that, they would use disincentives, and somebody would call it economic warfare I guess if they didn't like it, but much like words aren't violence, sanctions also aren't violence. They can have severe effects, but the difference is still important.
I believe that countries in the region simply want the Western way of life and protection, especially the countries that used to be part of the Eastern Bloc (largely involuntarily or with at least half of the population not being happy about it at all the whole time for almost five decades).
The western freedom and democracy style of government isn't without issues (see any criticism of capitalism), but for those who have seen the difference and can compare, or know someone personally who lived under Russian influence, the impacts of that choice on the Dollar or on American weapon sales really aren't a part of the consideration.
It may be what motivates the politicians, but as the example of Ukraine shows, a president can't just decide out of nowhere to make the country join the other club than the one that the people overall want to be in. Ukrainians have rejected Russia as it invaded them with what seemed like overwhelming force. I'm sure that any kind of pressure from the EU or the U.S. would be much more easily resisted by them, if they didn't want to be part of the western club.