> the ONLY way this ever takes off is if Github does it.
I disagree. What this need is a popular, well known and trustable organisation with enough money to invest into this. Github could be this organisation, but other could do that to, like Mozilla for example, or the FSF, Patreon, Facebook, Apple, maybe even google if they can sell a vision of them not shutting it down 6 monts later.
What is needed is awarness of such an option. Which would work better of course when integrated into the platform. But patreon seems to have proofen that it's also possible to do this with external services.
Also a trustable institution to handle the money
And finally a decent interface to manage your sponsorings, which also means someone doing the work in the background to figure out whether a project accepts donations and how. Would be easy if the platform supports it somewhat automatically, but could also be done manually for projects on other platforms.
I would even go a step further and not limit this to open source projects, but the whole web. Add a simple button into the browser to allow sponsoring for the actual visited url. With some background-service which is smart enough to figure out whether the visited url is a project on github/gitlab/sourceforge, or a blog, a podcast, some content-company , a streamer, or some dude on twitter with something you wanna support.
And if nothing can found to support, an issue is created and the humans in the background will figure out how to support what you consider as supportable, and will inform you if something was found. And with time some schemas and APIs will grow so platforms will allow automation fof this.