If Kane's desire is to win the UCL then he came very close to it in 2019 with the Liverpool v Spurs 2019 UCL final. With the age that Kane is I do not think he is going to be able to do it with Spurs again. Will he get the chance to win the PL with Spurs before his age catches up with him? Again I doubt it. Would he be worried that his England place is in jeopardy is he does not play in the UCL? very possible.
My notion of why players want to play for a team that is in the UCL is not so they can win the UCL, although that would be a bonus, but in my opinion it is so they can protect their national team place because many national team managers of European teams use the UCL as a benchmark for grading how good a player is due to them playing against the top teams in Europe. Therefore, because Spurs is not in the UCL, and the fact Kane is not getting any younger, is Kane thinking that if he is not playing against the best players in Europe, is the England manager Southgate going to drop him for someone that is younger and playing in the UCL?
Also with the numerous sponsorships that players have, how many of those sponsorships have clauses that say the player must be playing in the UCL and/or must be a continued member of the England squad because being a current member of the national team has clout and status. It is much easier to sponsor a product saying '..and with Spurs and England squad member Harry Kane' rather than with '....and with Spurs member Harry Kane'. Sponsors can be brutal with their requirements when wanting to sponsor someone.