He does, but nowhere near as much for Spurs. You could argue we play easier opposition, but he was the same against Spain, regards the playmaking striker role. I personally dont think he suits the diamond formation and is much better up top on his own, preferably in a 433.
The only player the diamond actually benefits is Son, who plays on the shoulder of the last defender. It's a disgusting formation for most teams, including this Spurs team, and Poch is correctly receiving difficult questions over his long-term persistence with it.
The fact that Kane still has more than twice as many goals as anyone else at Spurs this season - including Son, who should theoretically be standing out in the current situation - says a lot about the standards set by the main man. This is what a crisis of form for Kane looks like.
And, yeah, you're absolutely right to point out that he can do this stuff against elite opposition like Spain too. Until we retreated to hold the win, he took them apart in that NL game.
The paradox (or dilemma) of Kane is that, outside of the very best teams in the world, he's probably going to be your best playmaker as well as your best finisher (he is in his current squads) so ideally you want Kane creating for Kane. But since that's impossible, both Southgate and Poch sometimes seem to want to "split the difference" and try to get him doing everything. The problem is that instead of harnessing his elite talents in either area, they end up getting lesser output from him in both, and tiring the hell out of him in the process (not to mention making him look bad compared to the players around him who benefit).
It's a problem of coaching and personnel, like not having anyone else to do the other role at the same level (particularly a problem at Spurs right now). The blame rests with the people responsible for those things. It's not Kane's fault if he gets frustrated with no service and takes it upon himself to drop progressively deeper as needed to get the ball rolling (pun?). That's just doing your job as best you can.
On a sidenote, Kane and Sterling in a club team together would be outrageous. Their attributes are almost perfectly complementary and they clearly have great chemistry (arguably better than the chemistry between Kane and most of his clubmates at present).
