The person who posed the question regarding skiing commented that bike riding is "accepted by all." I would argue, not so, although I have observed over the years that many have become quite meiqel regarding these activities including ball playing...even in Lakewood to my surprise. I was quite shocked to see bike riding and ball-playing on shabbat upon moving to Lakewood recently. My son discourages my grandkids from this. It isn’t necessary. This also seems to be dependent upon not only Pesaq Halachah but also Minhag HaMaqom. When my kids were growing up in Baltimore, children did NOT play ball or ride bicycles. Even the little ones were discouraged from the use of tricycles. There is so much one could do to enhance Shabbat rather than looking for Heterim for activities which on their face are Uvdah Dechol, even if not specifically addressed by earlier generations of Chazal (i.e., bike riding, although ball playing has been addressed.) Not only are there issues of Uvdah dechol, but also bechuqoteichem lo teileichu, etc. Ball playing was not an activity of frum Jewish children in general, but neither was attendence at theaters, sporting events, etc. I guess this makes me a "frumie." Over the years, I have come to understand the wisdom of Chazal much more. When I was younger, I too might have thought, "What is the harm?"
ב"ה Shalom I'm not sure I know what your question is, but if I understand correctly the issue at hand is one of attitude and education of the value of Shabbat. If I missed the point, you can ask again . Each Shabbat and holiday we have is not the subtotal of Halachic restrictions and Heterim we have. Every Shabbat and holiday have an intrinsic holiness which fills the day, the atmosphere and our Neshama. In order to connect to the unique holiness and spiritually uplifting qualities of the special days which Hashem has given us, we have to tune in to their message. If we focus on the wrong thing, we miss the gift of these days even on the simplest level., and the harm form missing it could be terrible. Take for example, the Nine days preceeding Tishah B'av , in which, Ashkenazim are restricted from eating meat, poultry or drinking wine in commemoration of the destruction of the Bet Hamikdash. How much do we focus on "how are we are going to survive on fish, soya and tofu for 9 days? When can we go swimming again? And how much time do we focus on where would I have been or what I would have done to improve myself, if I had been in the time of the destruction of the Bet Hamikdash? What can I do today to bring us one step closer to building the Bet Hamikdash? The same issue of attitude applies to Pessach which is now coming up? Do we relate only to gastronomic issues of how our stomach will relate to Matza for 7 or 8 days (In Israel or abroad) or do we see ourselves as ones who have left Egypt and we are now free men longing to fulfill our mission in the world as part of the Jewish people. In a totally different context, but which now applies to everybody, what do we with our days of quarantine, kids not going to school ,not going to shul during this corona pandemic? What do we do with our time? Are we doing creative things trying to make the best of it for our family and ourselves or our we just texting all day on the phone and checking the latest corona figures or jokes? We can now see how all this applies to Shabbat observance. Do we just look at Shabbat as a day which confines me to no travel, no use of electronics, no phone (Cell or landline) ? Or do we look at Shabbat first on the simplest level as a gift that one day a week, I can disconnect from my work, my cellphone and be home to be with my family and friends. From there, we can take Shabbat to higher levels of Torah learning and absorbing the holiness of Shabbat. Once we understand the holiness if Shabbat and exemplify it to our children we can then discuss what games we can allow our children to do when they are a yet at an age to appreciate Shabbat the way adults do. We will then look in our books of Halacha, and learn about the restrictions Chazal made to preserve the special character and atmosphere of Shabbat, among the very popular ones are Shemirat Shabbat Kehilchata or the more recent, Peninei Halach , and see exactly which games our allowed for kids on Shabbat and which are forbidden. All the best