No he doesn't.I mean, he has the final word on who he gets, no manager would work under the terms you've laid out. He literally said himself that he and the club were slow on signing players because they working together to identify the right players.
Some nonsense gets said on here.
No Spurs managers have much of a say in the signings. Conte and Jose both said as much. If the highly decorated Conte couldn't get the signings he wanted what chance has Frank?
You understand that there is a world of difference between selecting from a pool of players that want to come and are in budget and in the right wage bracket vs getting your first choice to improve the team if money was no object right?