I can't speak for other parts of the country, but here in a trendy (read: HCOL) part of California renting stopped being trendy about 5 years ago. During the housing crash local media was full of accounts of smug renters going on and on about how renting was so much smarter than owning. Even coined a term for it, the "rental economy" - everyone was going to rent everything, and renters would grow wealthier as they saved money and owners shouldered the burdens of ownership.
That was the theory at least. In reality, those folks missed a once in a lifetime opportunity to buy in a HCOL area at fire sale prices. I tend to be somewhat of a contrarian, so we doubled down and bought rental property for cheap back then. Now renters bitch incessantly about the unfairness of increasing rents, and disparage "greedy" landlords for charging market rates. No one here talks about wanting to rent and we have people begging to pay more than twice what we paid for our primary residence, a crappy track home on a tiny tiny lot. So being the contrarian that I am, we're going to sell. For about one-third of what our house is worth we can buy a much larger house with an actual yard in a MCOL area.
If it's half as expensive to rent than own in a LCOL area then it makes sense to keep renting. But as the rent vs. buy gap narrows the intangibles start to become more important, especially for families. It's super easy to move when young and single: borrow your friend's pickup truck, and make one or two trips to move your studio or one bedroom. But moving with kids is a total pain in the ass. Lots more stuff. Changing schools. Disrupting kid's circle of friends. Just a lot more work. And as families tend to value a place they can more easily "make their own" so to speak. Don't like the layout or some awkward think about the house, well, can usually change that as a homeowner. Not as much with a rental.
So I would expect Millennials to become homeowners as they start having kids, especially in areas where the difference between rent vs. own isn't too great.
What does this mean? I don't think there's a general answer to this as every location is different. In "up and coming" areas and/or places ripe for gentrification it may be better to buy sooner rather than later. However this depends completely on the specific housing market in question.