Bell Labs could do this because their parent company was a government sanctioned monopoly that didn't have to worry about about mundane problems like quarterly earnings. Xerox had the copier machine market pretty well to itself for a good 20 years which enabled them to fund PARC (sure, there were some competitors in that time, but they weren't taking a huge bite out of Xerox's profits yet).

I think Apple could've done this a few years back when they had something like $100B cash on hand - they could have established a research foundation and provided it with funding that would've lasted for long enough for it to become self sustaining. Maybe Apple still could do this as they've got $73B cash on hand. $10B would go a long ways towards establishing an institution like this. Even better if other FAANG companies joined by providing seed funding, but maybe that looks too anti-competitive these days? (the incentive for providing initial funding could be a stipulation that funding companies get a break on IP license fees)

Of course, as someone else mentions in this thread, this institution should learn from PARC's mistakes and do a better job of licensing their IP in order to keep operating for a longer term.

I suspect we really need something like this. Bell Labs (and PARC) gave our economy a technological boost that it wouldn't have had without them. Maybe a revitalized SRI (with some additional outside funding) could become what you're talking about?