... but support for existing things get increasingly pushed onto support teams.
And support teams don't fix bugs?
You're removing autonomy from the support team, this will demoralize them.
The issue becomes, you have two teams, one moving fast, adding new features, often nonsensical to the support team, and the second one cleaning up afterward. Being in clean-up crew ain't fun at all.
This builds up resentment, i.e. "Why are they doing this?".
EDIT: If you make it so support team approval is necessary for feature team, you'll remove autonomy from feature team, causing resentment in their ranks (i.e. "Why are they slowing us down? We need this to hit our KPIs!").
On top of that support team often undeerstaffed and overloaded while feature pushers get more positions.
Some 20+ years ago we solved this by leapfrogging.
Team A does majority of new features in major release N.
Team B for N+1.
Team A for N+2.
Team A maintains N until N+1 ships.
Team B maintains N+1 until N+2 ships.