True, we do tend to cut back on spending after we reach the Champion's League positions (so long as we don't actually sign Sancho; it's looking like it's either him or nobody.), but I would trust Pochettino to hold us in a top four position without signings more than most managers. After all, he did the same thing with Tottenham in 18/19 (no signings, but reached CL final and top four), albeit, it did collapse after that.
I think if he manages that, it will be far more likely he will get the funding in the following transfer window after the summer of little spending (we always spend something, to be fair, unlike Tottenham - seems about £40m - £50m).
Pochettino is more suited to us because of his ability to get the most out of his players. With that, I think he could make better use of the windows where we spend less as well.
I think Pochettino can get better use out of some of our players e.g. Fred, Shaw (He did manage him at Southampton, so it may have some bearing) etc. This will help us with delaying certain areas of recruitment.
He will suit us more than any other manager and is the least likely to be in such a position to be thrown out. After all, it isn't only the owners fault we have been poor.