Harry Kane is playing for Tottenham for years and he is a famous player there, he is loyal to his team but because of the bad results he got in Tottenham, he is better to leave and find a team where he can achieve his goals.
Tottenham Hotspur's main issue is winning trophies; they will be at their best and still fail to win a single prize. Even the club's past managers, Anthonio Conte and Jose Mourinho, failed, so I can't think of any coach that is capable of saving the club because it's apparent that the problem stems from the boards and the players. I don't see any compelling reason why Tottenham Hotspur should have one of the best strikers in the English Premier League and still struggle to win a trophy. Harry Kane is the club's all-time leading scorer, and while he has not made his personal objectives public, I believe it is high time he tries a new chapter somewhere else.
Overall, Tottenham are a good team, it's just that as you say they can't win trophies, that's where they are in trouble. I don't see them being very unlucky, but indeed the level of competition they have shown has not been able to make them win trophies. they should be able to improve themselves from now on, otherwise they would never gain anything.