Yes, I suppose you could do it on the weekends. In our area (NYC/Long Island), I have never seen that done. It may be more common elsewhere. I think a Sunday School would be a good idea, followed immediately by Mass.
You are correct when you say that it is ultimately the parents’ responsibility to have their children confirmed (or to receive any of the sacraments). Sadly, many parents will not do so if it interferes with schoolwork, the parents’ work schedules, the work schedules of the kids themselves, or some other activity (usually sports).
Scheduling Confirmation for sophomores or juniors in high school only exacerbates the problem, IMO. I’m not sure why some dioceses decided to make it so late. Again, in our area, Confirmation is usually scheduled for anywhere from the 6th grade to the 8th.
I’m glad you were able to receive Confirmation! So many young people never return. God bless you!
Regards,