To quote your example, who have Spurs been trying to sign, but has turned them down because they're afraid to have the challenge of dethroning Kane as the number 1 striker?
Do you think that would've happened at Man Utd in the 90's and 2000's under Ferguson? Top players went there for the challenge, and if they were better than what he already had, he wouldn't think twice about replacing the existing player, no matter who they were. That's what made him, the players, and Man Utd serial winners.
Call me old fashioned, but I'm of the opinion that no player is untouchable, whoever it is. If he's not cutting it, he's coming out of the side, and it's up to him to win his place back, and I would assure any potential signing of my stance on that, and if that player were to knuckle down, and outperform the so called "invincible" player in training, then he'd get the nod on matchday, equally I'd put a rocket up the current "invincible" player to up his game even more if he wants to keep his place.
Great managers are not afraid to make those calls, and great players are not put off by competition and the possibility of missing out on a starting place, it's what drives them to be better players, and become winners.