If you need to "count how many times he is good this and that", that coach is already not good enough. To win the league next year, you need to collect over 90 points. To secure CL, you need 78 or so. You need to get your decision right most of the time.
If the hots balance the colds, Inter will be exactly where Lazio is today i.e. midtable.
Let's compare with whom he will compete. Is Inzaghi better than Allegri (or any Juve new coach)? NO. Gasperini? NO. Pioli? NO. Mourinho? NO. Spalletti? NO. See that again, your list of "Inzaghi achievements" have been done better by the others.
I'm by no means a fan of Simone Inzaghi, neither as player nor as coach, but... come on. The guy is not some clown, he's a decent coach. He's just whiny and arrogant, just like most coaches are. I think most people dislike him as a choice due to personality.
Allegri is a cunt, but he's easy going usually.
Conte is an asshole, but he's a charismatic one.
Gasperini is not a pleasant person.
Spalletti is actually the most fun character of them all (Italians I mean, Mourinho is on a different level than all these combined) and we all saw what mark he left.
Pioli is just happy to be around, but he's no master coach. Milan needs to move on from him if they want to return to their 2000s life.
Gasperini and Spalletti are actually the only ones that performed great with teams that no one expected to do very well.
Allegri was good at Cagliari, but he had a very nice roster and got lucky that Zlatan's Milan came knocking on his door.
Conte was piss poor everywhere he went [in Serie A] before Agnelli decided it was time to spend money on players and refs.
Pioli was a mid-table coach at best who can sometimes get the most out of his unit. But this season was the only one where this was sustained. I put more on this on the Zlatan in the locker room effect, than Pioli masterclass.
Simone Inzaghi only coached Lazio and he's been there for 4 seasons.
They were barely top 10 when he was hired in 2016. Sure, they did have a season where they finished 3rd and 4th, but they were typically a top 8 club before him. They had a higher budget than both Napoli at the time. This changed by the time Inzaghi came about. Their budget went from 70m to 30m from 2014-15 (3rd place) to 2016-17 which was his first season. From 8th in 2016, which he also managed a few games at the end, he took them 5th with the 6th highest budget (Atalanta also had much less and finished above them of course). But from 54 points to 70 points, it's quite a good increase.
They started to raise the budget since then but he was always operating with the 6-7th budget in Serie A.
70 points, 72 points, 59 points* (2018-19, but won the Coppa), 78 points, 68 points (and past the CL groups in their first try).
*Lotito masterclass managed to lose De Vrij on a free, sell Felipe Anderson, meaning that 2 of their top5 players were out and not really adequately replaced. Correa took a while to settle in a new role.
And all this, while having an ultra thin roster. We've had similar stories with Spalletti at Inter. I'm expecting both to be able to bring far better results for a better team with more depth. And we're one of those now. But Spalletti cannot return, that ship has sailed.
This coach would not be a bad option, in fact, it's one of the safest ones we can make. Is he great? Of course not, but pretending that we're losing a great coach here is also silly. The main problem is that neither Allegri nor Sarri seem likely and there's really no one else to look at. Even Pochettino may leave PSG but I still wouldn't claim he's an exceptional coach that we must have.
I'd prefer Allegri from the lot, but Simone Inzaghi is part of the "next best thing" whether we like it or not.