Due to work in a restaurant, and it being extremely busy, sometimes I miss either Asr or Maghrib, I’m not sure if this counts as a valid excuse, what should I do in this case? Should I make it up later?

Work is not a valid excuse to delay a prayer past its time. The five prayers have fixed times, as Allah says, the meaning of which is: (indeed the prayer is upon the believers a decree of fixed times) (al Nisa 103), and a prayer takes only a few minutes, so you pray it within its time even while at work. There is always a way in sha'a Allah, so try to find a way to pray or you can be on the lookout for a job that does not prevent you from salah.