I don't think Tottenham have really inspired any more optimism from Kane's perspective so far this off-season. The coaching search was a complete farce (although some of that was probably due to poor media reporting) and the team doesn't feel like they are in any more of a competitive position then they were at the end of last season. I think it's pretty simple; Kane wants to be in a position to win things and the team over the past two years has not been in that position, and Kane is looking at his prime being wasted on a team languishing in the middle of the table, even if he leads the league in goals and assists. He was Jose's biggest ally on the team and I think that was because he felt like Jose understood how to win and he didn't think the rest of the organization wanted to do the work necessary to win (Kane is probably wrong in this instance, Jose was running an outdated system and had fallen out of touch with the game).
If Tottenham make some really positive moves, sells some guys and brings in some high-level talent, and Kane has some good conversations with Nuno, it could all be smoothed over, but I don't see things trending in that direction. This feels like a full squad rebuild and at that point, keeping Kane around makes little sense.
The mystery in all of this is that Kane doesn't seem to have a landing spot if City are sincere in not being interested. Kane is serious about not wanting to leave England, with a key reason imo being that he really wants Shearer's record. Man U are out of the running after the Sancho move. Chelsea could always pony up the dough, but that would require a few other moves to be made, such as moving on from Werner and that doesn't seem likely. Liverpool and Arsenal won't pay that much for him.