Na, I don't think Russia played into this all that much. Excessive ambition, overzealous idealism, and internal politics were much more relevant. Don't forget that when these decisions were made Russia wasn't perceived to be the threat it's seen as today, at least not by the EU of that day. E.g. poland joined in 2004, but much of the talks were in the 90s- and thus in an era when Russia was far from threatening. Putin didn't become president until 2000. So while the EU certainly wasn't looking for a buffer, it's much more plausible that the new members were less sanguine about future Russia (and how right they were), thus seeking to cement their independence from the USSR.

It's somewhat ironic, for instance, to consider that it was the UK that particularly pushed for expansion. Not only did they lobby hard internally beforehand, they were (IIRC?) only EU country to immediately grant free immigration from the new members, without any transition period - how times change!