I think I would pick Amorim if I were a Spurs fan. But Nagelsmann and De Zerbi would also be good choices. Realistically, I think Spurs are entering a rebuilding stage and need like a 3-4 year plan to turn over the side and put together a new team. Amorim has fresh ideas tactically, has a track record of working with and developing young players, also managed to slay some dragons and actually win Sporting's first league title in 20 years, and would see Spurs as a clear step up and likely be in for a lengthy project. De Zerbi is similar but Amorim has accomplished more IMO. Nagelsmann is a good young manager but I don't see him as the right fit for a 3-4 year project, likely involving some lean years at the beginning. He has never really committed to any club for very long and having tasted life at Bayern, and probably feeling that his time there came to an unfairly premature end, I don't see him being committed to carrying through a long rebuilding project elsewhere that might involve a multiyear stretch of foundering around 5-7th place and being out of the Champions League.
I wouldn't want Poch, too much history there and I don't think he is particularly strong tactically in terms of what his teams do in possession. I rate Luis Enrique highly but he has been out of club football for so long that would scare me. There really aren't many examples of managers taking a six year hiatus from club football, then coming back and doing it again at a high level. The only one I can think of that even tried something similar was Sacchi and that ended very badly.