No, non profits can invest in anything. Publicly traded stocks are c-corps too, thats how endowments grow. There is nothing that distinguishes liquid vs illiquid c-corp shares.

Regarding founder ownership, the rules are extremely flexible like a non profit director can’t own more than 20 voting or 35% total of the business venture

but if it happens then it just needs to be remedied within 3 years

so for venture style deals that’s plenty of time to dilute down, and the little known secret in the startup space is that the founders non profit steps in as the lead investor, so all the other investors arent just twiddling their thumbs waiting for a founder to convince someone, it just closes. Other investors dilute founder and non profit, everything is compliant, value is created. Both for profit and non profit side will be tax free, due to QSBS