I don't think your either/or options are totally accurate.
- House vs apartment. Have you actually looked at the costs to buy and rent a (large enough) apartment? Including HOA fees (house and apt), taxes, upkeep etc. Sure everyone don't need a room, but family of 5 would still mean a sizable apt.. At least in this area renting is barely cheaper than my full mortgage, if at all. Friends renting a townhouse actually pay more rent than we do for a SFH mortgage! And we have high taxes and HOA fees. Of course we have upkeep, but also some appreciation and equity. I don't think your math is as cut and dry as "$250k more". Some areas it is, others not so much.
- You seem to contrast "play in yard" vs "pay for experiences". OK sure, but I'd argue not totally equal. For one the time. We're busy, so sometimes the kids play outside a few minutes while we make dinner etc. You couldn't really go to the zoo in that time.. You could have a park nearby, but that's still a bigger commitment than letting the kids run outside in a (fenced) yard. And of course you could have a nice playspace outside your apartment as well, so it's not like a house is the only way to have that.
- other benefits of house. House vs apt are very different, than just yard vs money. You probably have more space. For example allows me to easily do woodworking, bike/car maintenance etc. Or any other crafts, toys like kaykas, tents etc (i.e. cheap, wholesome family activities). We work out in the basement, saving time/money on gym. We do our own upgrade and DIY, having fun, learning skills and adding value to our house. You can ofc do this in a apt, but you're more limited.
I see the benefits of both. They are very different. I'd say choose which lifestyle you prefer. Not based on money (which might not actually be a big difference anyway)