His football is far from inspiring. I wouldn't want him at Utd, but I suppose for Spurs to have a manager of his pedigree would be pretty massive for them.
Ok, inspiring is maybe not the first word coming to mind, he's indeed not a very expansive or attacking, but a properly pragmatic manager. If Bayern was looking for a manager, it would maybe be a similar deliberation to yours, and maybe I'd prefer someone like ten Haag over Allegri based on style and approach.
However, in the end, you've got to come back to quality, and Allegri's record is impressive and he is undeniable top quality as a manager.
He sets up teams to win, and I've repeatedly been impressed by his ability to change a game through subs and tactical rearrangements. I remember our game at the Juventus stadium in 2016, or I remember the CL tie Juventus against Spurs the other year. In both games Juve started poorly and were outperformed (of course you could say that's also his fault), but he made changes to turn it around very clearly. He improves players, too.
Won the league every year, two CL finals, and also managed to win the league with a Milan team, back then, very much already in decline, even if fans and board didn't want to realize it yet.
And especially following after José, it would make sense, as the squad now has started to be formed for Mourinho - and Allegri is the kind of manager in reality that Mourinho thinks of himself he is, but hasn't been in a long time. A consistent, pragmatic, tactically expert winner.
Would be a coup for Spurs, really.
I could imagine though that they just can't afford to sack Mourinho, financially.