I don't get it. Why use a preprint server associated with publishers, when publishers have the history they have? What's the difference with putting them on e.g. arxiv?

Arxiv has a narrow focus (only physics) and it's rather difficult to even get permission to publish there in the first place. Cold Spring Harbor Laboratory runs BioRxiv and MedRxiv for biology and medicine, respectively, but again there's the issue of focus. Those are the big three. Then IEEE started TechRxiv as a preprint server for tech/engineering, which, given the landscape at the time, was a good move.

Arxiv isn't just Physics at all, historically it's always been a big place for Maths and CS too.

Yeah, that's very true.

It's very easy to get access to arxiv.org, if you're at a university or a company doing research you'll surely have a colleague to vouch for you. Even if you're a lone wolf, just ask someone on a subject-focussed site or email a local professor.

*Disclosure: I run a preprint server that "competes" with TechRxiv at https://engrxiv.org.

I'm a little salty about TechRxiv/IEEE as IEEE has a copyright transfer policy that permits posting preprints at approved servers. I applied to have engrXiv recognized as an approved server but received no response. Shortly after that TechRxiv was launched. Currently the only approved servers are still ArXiv and TechRxiv.

I'll go try to answer these for myself, but — using the "both things can be true" argument — I have a couple if follow-up questions:

1. Do they still require transfer of copyright to IEEE?

2. Or, conversely, do they publish public domain articles?

1. No. "TechRxiv offers several Creative Commons licenses, all of which permit you to retain copyright of your work. You do not need to transfer your copyright to TechRxiv."

2. Preprint servers aren't really for the publication of articles that were already published elsewhere. But you can have an article published on TechRxiv peer-reviewed and subsequently published in an IEEE journal -- or a non-IEEE journal, for that matter.