Arne Slot has good start with Liverpool but to win Premier League when Manchester City have been dominating it recent seasons, is not easy. There is another competitive team, Arsenal, that are ready to beat Manchester City in this season. Liverpool and Slot will have to resolve match by match, aim at maximize results and after 10 to 15 matches, they can have better overview on their competitiveness for the title.
The victory obtained by Liverpool is certainly a good thing for them but bad for Manchester United, so they can be in second place in the standings. It's true what you said Arne slot started the season well with Liverpool and I think Liverpool is ready to face a strong Manchester City, with them being able to beat the red devils this is a good achievement but they can't assume Manchester City is easy to beat so they still have to be able to prepare well.
It seems that Manchester United still has a problem because with this match I saw how freely Liverpool played the ball without the tight supervision of Manchester United, I think if they still don't develop maybe Ten Hag will end.