Running around without the ball brings you the ball because you win it back when you lose it. If you don't work hard to win the ball, you won't have the ball - such is the Premier League.
Top 5 Possession in 2021/22:
1. Manchester City
2. Liverpool
3. Chelsea
4. Brighton & Hove Albion
5. Arsenal
One of these teams operated on the second lowest budget in the league, the others didn't.
De Zerbi's Brighton and Potter's Brighton are very similar stories in terms of tactics and approach - hence the smooth transition. Short passing through pressure which creates spaces on the opposing half where the team overloads in one area or another to create chances.
Manchester United a year ago had no more fun than you had this weekend:
https://www.youtube.com/watch?v=i7m-Xi8pYIc
Also I don't understand why someone would "fit Spurs" but "not Chelsea" when you have been twins for about twenty years, (mainly) employing defensive, pragmatic coaches -in a lot of cases the same ones - like Scolari, Mourinho, Conte.
You're right that GP, who is known to have his clubs play brave and attacking football, doesn't fit into that. But he'd have the same problem in Spurs: incompetence in the board room, a club identity of playing powerful rather than fluid football, and a bunch of overpaid divas unwilling to put the amount of effort and work required to reach the top on a consistent basis.
Largely you can divide football managers into two categories: Cruijffians and Gaalians. The former are utopians who want their teams to play sexy football. In this category you'll find Cruijff, Pep, De Zerbi, Potter, Wenger, Roberto Martinez and of course dozens and dozens of others. The latter are "win at all costs"-managers like van Gaal, Mourinho, Ferguson, Benitez, Simeone, Scolari, Zidane, Capello, Conte and, again, dozens and dozens of others. Chelsea has been a Gaalian team since the Abramovich era began, with a couple of exceptions like Sarri (mainly a Cruijffian) and Guus Hiddink (one of few successful 'hybrid' managers).
These approaches doesn't generate different results really. There's advantages and disadvantages to it.
The Gaalian approach requires experience, muscle and grit, and the advantage with it is that if you have experienced, good players, you can get them to play that way within months. Another advantage is that results will always be at least 'ok'. The disadvantage is that since you're relying on experienced players, it is difficult to keep the level up for years, and it can be very difficult to introduce young players into it.
The Cruijffian approach requires technique, a willingness to participate in all areas of the game, a desire to learn and full respect for your teammates. The advantage is that the end result is great to watch and can last for a very long time since the players learn to play according to the system rather than the system being adapted to get the most out of what already is. The disadvantage is that it takes a lot of time to make it work and when it isn't working, it looks... really bad, like in Frank Rijkaards first Barcelona season when he was laying the foundations to the team that would eventually dominate under Pep & Enrique.
Spurs have a reputation of being a Cruijffian club but they haven't been for (at least) twenty years. Chelsea wanted to be a Cruijffian club in the 90s and early 00s but has been Gaalian ever since. Todd Boehly wants - or wanted - to change it, apparently, and Graham is the right man for that. But you can't build a Cruijjfian style if you sack the manager after six months of poor results. Tony Bloom knows that, Todd Boehly doesn't. Too bad for you.