The (misguided IMO) idea was that buying their gas and integrating them into world markets would strengthen ties and liberalize them in the medium term.
I don't think it was misguided; Nuland-Pyatt leak and the tapping of Merkel's phone made it pretty clear to me that like in the Yeltsin years the problem was that the US didn't want Russia/Europe ties to succeed.
Why then did Russia have $300 Billion invested all over Europe? That was a massive portion of their national wealth, and it all got frozen by the EU in 2022. It looks as if Russia did want close economic integration with Europe. That's something the usa would not want, though.
Russia invaded Ukraine specifically to stop it from becoming too integrated with Europe. The idea of a rich and democratic Ukraine is a nightmare for Putin.
Nobody believes that anymore, post-2022.