This is very good to know! I am currently setting up my old pixel 6 to be a permanent Google home terminal. I was thinking about trying Lawnchair, now I'm not so sure.
To be fair, this was probably six or so months ago when I last went through this. Niagra had the launcher / app switching issue, so did Lawnchair, even Nova Launcher (before I knew of what had happened to them). It could have been a general bug for all launchers except the OEM.
Haven't seen the app switching issues with Lawnchair for a long time, both on pixel 6 and pixel 9. So I think this might finally be resolved..