I would think most able DCs take 5 Highers in S5 and then 2 or 3 (or occasionally 4 if aiming high) AHs in S6. It is possible to go to a Scottish Uni straight after Highers (except for Medicine or Vet Med), but I think it is generally preferred to have studied to AH level, and you'd need AHs to apply to English Unis (if you were going down that route). I don't know any of DS's cohort who didn't stay on for AHs, and your DC may not want to go to Uni at 17 and be unable to go to bars and clubs with their mates. Maybe also useful to be know that there is no AH equivalent of Further Maths.
S5 teaching will start after the end of S4 exams, so you could be missing the first 4 weeks or so of the syllabus. Given that you can take crash Highers in some subjects (so Highers where you haven't previously studied the subject) I don't think you'd need to worry about playing catch up too much.
Also be aware that you likely won't have the required 3 years residence for the free fees, although £1820 per year is still way better than the English fees.
A big plus point is that if you apply to Scottish Unis at the beginning of S6 then it is on the basis of known Higher results, not sure if things are now moving more to conditional offers but DS got all unconditional offers for his engineering course a couple of years back.
For our school, local admissions were generally guaranteed if you lived in the catchment, but you had to have an address in the catchment to be able to apply (popular school). Best to speak to the local authority maybe?