While wanting to build an open-source alternative to a shit platform run by an evil company is a nice (even admirable) thought, unless you really think on the myriad 'why's and 'how's and come up with some sort very-high-level spec that's more than 2 sentences long, this will most likely remain a thought. And not only that, a rather unoriginal one since I'm pretty sure there is at least 1 open-source LinkedIn replacement out there (checking https://github.com/awesome-selfhosted/awesome-selfhosted is usually a good starting point considering your goal of wanting co-ops to run their own instances). If there isn't (or the others are just bad for whatever reasons) then maybe the community can start having real discussions for this.
If however your intention for posting this was to try and get work to organically start on an OSS LinkedIn clone started, then good luck. But "let's make an OSS version of X/Y/Z" is basically the same as "let's buy a bar" or "let's start a band", seldom do these thoughts lead to their desired outcomes.