The fact that there are so few of these types of company around probably means that they have trouble surviving in a world where the competition is supported by modern finance.
The problem with coöperatives is longevity. We have white-collar coöps of sorts: partnerships. They tend to disintegrate after a generation or become quasi-corporations over time because immortal entities can make plans and promises on time scales that ones that grow old and change priorities and die can't.
The John Lewis Partnership in the UK is a public limited company whose shares are entirely owned by a trust that pays shares of profits to all of its employees, which isn't quite "ownership" but close, and they have a $15bn revenue. Publix Super Markets in the US has a $45bn revenue and is privately owned with all employees receiving stock.
https://www.investopedia.com/articles/insights/051316/6-succ...
Financing is probably an issue at a certain level but it doesn't seem to be limiting otherwise. They're just unusual, although if you consider startups giving employees a stake isn't that strange today.
Also, LLP is a limited liability partnership. Wholly state entity, though, so the same caveat again. Member limits, etc.
And that's the USA of course. Your country may vary.
Yes, in the US, at least. A coop can be a corporation or LLC (and a coop that is either the former or the latter when taxed as a corporation has special tax status with the IRS, under subchapter T, so occasionally they are referred to as “T corps” analogous to C or S corps.)