You seem to not know what we don't know which seems perfectly logical.
One thing I can assure you of is that not many people born in and around London (like Kane was) aspire to live in and around Manchester. In that respect cities are a bit like football clubs, some are aspirational and attract the cleverest, most talented and wealthiest people from all walks of life and from all over the world, and others don't, meaning Kane is probably very happy in London.
I'm not saying they aspire to live in any location. A footballer growing up aspires to win as many trophies as possible and earn as much money as possible. Location is irrelevant as evidenced by pretty much every player choosing to earn the most they possible can whilst competing for trophies. Again I can give numerous examples of players choosing Man Utd or City over a London club. Give me one example a top player choosing a lower salary in London over a higher salary in Manchester.
Plus we're no exactly talking about playing in China. Manchester to London in a helicopter isn't exactly a 12 hour journey.
So in the past few offseasons no one has wanted Kane, Lloris, Alderweireld, Alli, etc.?
How about this offseason? Everyone is going to pass up on a young english player who scores 20+ every year? When CBs are in such high demand none of them would bid massive amounts for Alderweireld or Vertonghen? They'd rather pay the 50+ for a Stones or Otamendi? Really?
Yourselves, City, Chelsea, etc. would jump at the chance to sign Kane and given money is no object to those clubs then the answer seems to be clear.
At the amount each player would cost at this moment I don't believe any of those 4 players have attracted significant wage offers from elsewhere. Of course people would be all over them if they were being sold at £10m, but that would never be the case. In terms of Lloris/Alderweireld I can't see who would want to spend £40+m and £150k a week on either player, City aside. I agree City would have been better off, but they wanted a different profile of signing.
In terms of Alli/Kane who would want to spend £75-120m and £200k a week on players who are still in the infancy of their careers and who aren't proven at the highest level.
Spurs' biggest attribute is that they can demand massive transfer fees and on most occasions teams will look go better value abroad. Why sign Alli or Kane for £75-120m when you can get Griezmann, Pogba or Ibrahimovich?
Naturally that starts to change when these players become the best in the world, they look at similar players' salaries and their contracts have 2-3 years left rather than 5.