It’s so important, in fact, that there should be more than one such institution.

People keep falling into the same trap. They love monopolies, then are shocked when those monopolies jerk them around.

I am using Zenodo for a while now instead. It is more user friendly, as well.

It is just a preprint repository. It is pretty open (the stories where a preprint was rejected or delayed unreasonably are extremely rare). It offers the basic services for a math/compsci/physics themed preprint repository.

I don't see much of a monopoly, nor any "moat" apart from it being recognised. You can already post preprints on a personal website or on github, and there are "alternatives" such as researchgate that can also host preprints, or zenodo. There are also some lesser known alternatives even. I do not see anything special in hosting preprints online apart from the convenience of being able to have a centralised place to place them and search for them (which you call "monopoly"). If anything, the recognisability and centrality of arxiv helped a lot the old, darker days to establish open access to papers. There was a time when many journals would not let you publish a preprint, or have all kinds of weird rules when you can and when you can't. Probably still to some degree.

there is. bioarxiv.