I’m not considering this point an obstacle because we should assume that any project that is critical and in a desperate state is willing to be funded. Even when the maintainer does not want to receive compensation for their time, having the ability to hire a trusted contributor or maintainer to take over when the volunteer is exhausted, burned-out or wants to give up their hobby. The willingness to receive funds for your own work does not mean you don’t need money for other purposes. As the XZ affair has shown, it is increasingly difficult to impossible for a maintainer to find a trusted maintainer to take over from them and if the previous maintainer has to suspend work on a project urgently, the only way to get a trustworthy maintainer in quickly and reliably is to hire them.
On the question of which institution should take over this role? I’m not sure. There are few foundations in the FOSS space that aren’t controversial in some way.
I believe that whatever institution is interested in doing something like this (e.g. EFF, FSF, CCC, NLnet) will do something positive with it. It would of course be preferable if there was in institution, like a Guild, that had the reputation amongst FOSS developers and was organized by them and could take charge of such an initiative.
See: socialcoding