I think you were the exception though; I've never seen a start-up up close that allows you to trade or even pledge your equity for any purposes.

Yes; the idea is that the startup wants control for who will be on the cap table, so they don't hassle with written requests for info etcetera. Minority shareholders (especially for California businesses) have a fair number of information rights, and so companies tread cautiously in terms of who they allow to invest. This is in part why special purposes vehicles exist so that there's a point of contact for investment. (And then the company doesn't have to vet people that invest in that syndication as much...)