It was a mistake for NATO to do it without Russia. It's not a matter of "influence", but a matter of military risks and perceived threats.
If there was a political will towards pushing Russia to NATO any similar to what has been going on with Georgia or Ukraine, we'd be living in a different world.
But now (really far more than a decade already) Russia is alienated from NATO, perceives it as a threat and a bad actor. And people will be making all kinds of moralistic arguments, but point is that all of this is a consequence of geopolitical natural laws. Fact is, politicians and leaders were clearly aware of those laws and all the cause-effects and general outcomes, but nevertheless had stirred the world towards it.
