Well, from our experience, weeks run like this:
Monday: home from school at 4pm. Eat, change, chill. Leave at 530 for training. Finish training at 730, eat dinner in car on way home. Home by 8. Shower, chill (for 13 year old) or study (for 15 year old)
Same on Tuesdays and Thursdays.
Wednesday and Friday eves free, tend to go to gym after school, meet friends/boyfriend in town, head home. At home have dinner, watch TV, study, all the usual. Tutor on Fridays for 15 year old.
Weekends out of season tend to be up and out by 073p both mornings, train for a few hours, home by lunch time, afternoons and evenings free. Probably one weekend in three with nothing on, the kids will do a gym session or run or something instead.
During the season, those weekend sessions are often swapped for an all day event on the Saturday, so Sunday free.
So yes, they're plenty busy, as are we by extension. But there is still plenty of time to relax, study, see friends and family.
My family see our kids in the week by offering to do one of the training drives, or coming by for tea/cake after school, or a takeaway on a Friday night, or Sunday roast or whatever. As such they have a close and supportive bond. The issue is, the OP's family don't want any slots other than the Saturday morning because they don't want to compromise on any others. So they're being just as, if not more, rigid than the OP is.
I'd say my kids have a far more balanced life than some, filled with family and friends , boyfriends, school life, plus a sport they love and have gained great fitness, success and discipline from. Isn't that what we all want for our kids? And yes, they seem to find plenty of time to doom scroll too 😂