I think he's shown that his Spurs team can play a sustainable, entertaining and long-term efficient style of football that could, with an elite level squad, challenge for the league eventually...unlike ETH for example, who just abandoned his philosophies in his first season after the first two games, and grinded out 3rd with a deep defensive line and a very pragmatic style of football...and when he tried to return to that original approach, we were one of the worst teams across the whole league in the underlying metrics. So in this regard I think Postecoglou has proven more than ETH, but their squad would have to slightly overperform in order to make top 4 or even 5th place IMO.
He also kind of oversold both himself and especially this Spurs squad last season with that great run of form in the first three months, which might've made people think that all is well and good and all Spurs needed was a good manager and they'll make top 4 without many issues starting from now on...
...but has there been a more difficult time to even become a mainstay top 4 team in the league, than present day?
I don't think Liverpool are the best team in the world or that they'll even win the league, but they have a great manager, and a much better squad that he can work with. Arteta's Arsenal are a top 3 team in the world with a very young core that hasn't even peaked. City have been the best team in the world for years now and their slump only started 1 month ago. United have just hired one of the most exciting young coaches across Europe and Ineos seem to have high ambitions and are willing to burn a lot of money + hire the best in class footballing people in order to return United to the top again. I'm not sure I can change my opinion on Chelsea and their project just yet, but they definitely have more money than Spurs as well, just like all the other 4 clubs I've just mentioned.
So, for Spurs to win the league, or at least seriously challenge for it, or become a solid top 4 team for several years in a row, it would have to be a big sporting success. Similar to Leverkusen's huge sporting success in Germany, without being one of the richest clubs there. There are at least 5 teams that have more resources, elite level managers, and I'd argue better squads as well, with many world class players that are simply way superior profiles to anyone on the market currently, let alone the ones Spurs could realistically snatch away from other clubs on the market, like we did with Yoro from Madrid. Spurs cannot buy anyone currently that's as good, or even have a similar ceiling as Trent, van Dijk, Alisson, Salah, Palmer, Haaland, Gvardiol, De Bruyne, Rice, Saliba, Saka, and I'll put Bruno from United here as well because he's the 2nd best #10 in the league.
Also, defeats like yesterday's vs Bournemouth can easily happen. Iraola is an excellent coach and they've assembled a very respectable squad (or at least starting XI) that have many good players who could play for a bigger club than Bournemouth. Spurs also had some very important players unavailable: Romero, van de Ven, Vicario.
I also feel like Postecoglou wasn't backed properly in the summer. Solanke was an addition who should be able to contribute massively from day one, but Gray, Bergvall, and Odobert are not ready to do that, despite all 3 being very talented players IMO. It's a bit similar to United's 2018 (Fred, Dalot, Grant) and 2020 (Telles, Cavani, VdB, Amad, Pellistri) windows IMO where I felt like both Mourinho and Solskjaer were massively fecked over by the Glazers.
Ange is definitely not faultless though and I feel like there have been occasions where he was way too stubborn and it hurt his team's prospects to get a result, like the 9v11 high line vs Chelsea a year ago. He can also be a bit of a prick in his press conferences.
Overall, though, I don't think Spurs should sack him, because of two reasons.
1. I don't think they can get an objectively better manager to replace him.
2. Spurs cannot be expected to have a better and more exciting footballing project right now, than Arsenal, Liverpool, City, United, or Chelsea.
Long-term, it's different, because yes, they have some elite pieces that are ready to contribute right now, but there aren't enough of them. Vicario, Porro, Romero, VdV, Maddison, Solanke, Kulusevski, Johnson, Son. These guys are their best players IMO, but the quality of teams in the top 6 is just way too strong for this core to be expected to make top 4-5 comfortably IMO.
However, they have an incredibly exciting group of talents that might not be ready now, but should be in 2-3 years' time: Moore, Gray, Bergvall, Sarr, Odobert, Lankshear.
So overall, I think the manager isn't the problem, and they should temper expectations until they actually have a squad that's capable of achieving the club's long-term ambitions. Whilst their long-term prospects are looking great, if they can keep 90% of their current core + talented youngsters, I don't think any of Pep, Amorim, Slot, Arteta, etc. could get top 4 with their current team. Or it would be an incredibly difficult task to do so and would need some rivals to have bad seasons as well.