It depends on the cooperative, but the most common one, is that the cooperative owns the building and only residents can own shares. Those shares are only for sale to future residents and can only be bought and sold from and to the cooperative. Hence no free trading. Generally for older ones they tend to be rather cheap.

So yes the building is somewhat owned by the residents, but in many cases it is deliberatly turned into an awfull asset, which is unprofitable for the owners.

Even when buying a house, you usually have to take a massive loan. Those take years to be paid back. Especially when you have to take them, before you actually have a high income. After all young adults do not tend to have high incomes.

Research in Canada found that housing cooperatives had residents rate themselves as having the highest quality of life and housing satisfaction of any housing organization in the city.[8] Other research among older residents from the rural United States found that those living in housing cooperatives felt much safer, independent, satisfied with life, had more friends, had more privacy, were healthier and had things repaired faster.[9] Australian researchers found that cooperative housing built stronger social networks and support, as well as better relationships with neighbours compared to other forms of housing.[10] They cost 14% less for residents and had lower rates of debt and vacancy. Other US research has found that housing cooperatives tended to have higher rates of building quality, building safety, feelings of security among residents, lower crime rates, stable access to housing and significantly lower costs compared to conventional housing.

https://en.wikipedia.org/wiki/Housing_cooperative#Research_on_housing_cooperatives

The residents own the building indirectly. That means all of them look out for it. Unlike a commercial rent, the people who own it also live in it, which means they actually care about the quality. The cooperative has an actual structure to organize repairs, upkeep of shared spaces like floors and a way to collect money to pay for those. A private residence especially the later part can be a problem. Not having the money to pay for repairs or upgrades is rather common, especially for the retired.

Not all decommodified / cooperatively owned housing needs to be the sort of social housing that tend to come to mind when thinking of a "housing co-op."

Check out the work that the Beverly Vermont Community Land Trust is doing in Los Angeles: https://laecovillage.org/community-land-trust/. The "eco village" operates in this more crunchy, housing co-op sort of way, but then there are also lots of tenants and home-owners alike who live on the land owned by the land trust, without owning their homes in the standard sense.

That's just not true. As long as you have a defined structure and responsibilities