Overall, I think Matthew Hutton has a fair case, and I'm sure an extra HS2 station (or several stations) would have plenty of users, but I still don't think it would make sense. The two big issues remain, as others have said, reducing capacity on HS2 and slowing down every other user (there are far more people who will be slowed down than benefit from the new station).
If Aylesbury was much bigger (the size of Stoke, say), or if it was 10 miles further north, and on the actual East-West rail line, then it might become worthwhile, but it isn't, so the connection possibilities are minimal. Aylesbury Vale Parkway only has a bit over 100k passengers. Yes, if the trains to London only took 15-20 minutes I'm sure the numbers would increase, but a lot would just be due to lower numbers at Aylesbury itself.
Having said that, Aylesbury is fairly poorly connected and the existing line should certainly be connected to East-West rail. If it was feasible to extend this to Brackley and Buckingham, on the old line to Banbury, I can see how there would be positives from that, as all three towns could then have connections north and south. The existing route should also be sped up, if possible. That's a slightly separate issue though.