They will just sit back and let their "new territories" become more and more Russified.

The majority people who live there and have always lived there are Russian

It's like a game of chess,

No it really isn't. Its a proxy war forced by US aggression (2014 coup and NATO encirlement). The US achieved its primary goals - 1) to stop its European vassals (primarily Germany) from gaining energy independence through trade with the Russian Federation and get them back in line 2) to sell old weapon stocks/war profiteering for defense industry and make room for new weapons stores/more profits for defense industry.

The fact they've met these goals is the main reason why funding will stop, because there is no reason to fund Ukraine anymore. The US has won - Ukraine was always going to lose in this scenario.