That ignores any interest on the savings. So yeah it seems like "small" regular spending in your 20s (e.g. $100/mo total) can add up to an entire down payment in your early 30s.
Alternatively, assuming your rent would be at least $1000/month, you might live with your parents for an extra 13 months (or fewer for higher rents) before moving out on your own.
Also, what would happen to people with not so well off parents?
If someone's parents were able to provide a place to live until 18, why is 19 suddenly impossible? You don't need to be well off to let your kids stay with you a little bit past legal adulthood. Surely there's room for a couch or a bean bag or something to give them a huge leg up right out of the gate, assuming you were in a marginal position where you otherwise needed to immediately kick them out and downsize.
Some people are going to have rough situations and can't rely on a family at all. Some will need to emancipate before reaching 18. How common is that? If it is common, that seems like the real problem to address.