I think it would be an unusual 4 year old doing lots of activities. Mine could not have coped with much at all at 4. My youngest is 3.5 and has no regular commitments as he's often too tired. I've never really come across preschoolers being over scheduled so your family may have v unrealistic expectations.
My older dc do loads of extra curricular stuff. They absolutely love everything they do and no way would I take them to things they were reluctant to participate in.
I think it's been positive all round - they have tried lots of different sports and activities and have a good working knowledge of a lot of things which will help them to pick them up at a later age if they'd like to. They have stuck with some things and become good at them, competing and being part of a team which has taught them the sense of bonding and comradeship you can get plus the highs of achieving and feeling proud of training hard to reach a goal.
They have made friends with lots of dc from lots of different schools and a variety of backgrounds and cultures, taking part in weekends away and residentials etc, broadening their experience and giving them independence in a safe way. Having friends and activities outside of school was a lifesaver when dd1 was being bullied and was v unhappy in a school.
Mine have loads of free play and downtime and I find it easier to keep them off screens if we have plans and places to be. They never seem to get tired and often have an hour or 2 of pretend play before school in the mornings for starters!
I'm really pleased that mine are into so many things as they approach adolescence - hopefully they will remain active, engaged and enthusiastic as teens. I love to see the older kids at my dc activities as I very much want mine to engage in the real world as they grow rather than retreat behind a phone as I see so many older kids do.