This isn't something I was aware of, and I'm not having much luck in finding out the implementation details.
I can imagine a zillion different approaches... most obvious (not necessarily the best) one being to ask an IANA server over a normal TLS connection whether it should expect a TLD's records to be signed. And you can obviously cache that response for a while.
Remember the point here is to validate domain ownership, which everybody already understands to be a big deal. If you can't get any trustable response from anybody, then I would expect it is your duty to refrain from issuing a certificate for that domain.
But the larger issue is that there is no global integrated WHOIS system – e.g. to view a WHOIS record for a .de currently you need to solve a captcha and provide a valid reason (and can’t do it automated).
DENIC has been compliant with the GDPR for months already, and its WHOIS database is still running – you just can’t expose all fields of the WHOIS to any unauthenticated viewer.
Not to mention that, more practically there's no need (and potentially possible harm) to tie the ownership records to a particular cipher. Really, it's the registrar's job to deal with domain ownership; ideally it shouldn't involve you at all. And as a near-corollary, they could probably deal with long-term key security better than the average domain owner.
Really? In your mental model is (or should be) every domain owner overflowing with money?
A single static landing page would be expected to have at least traffic tracking, at least for the sake of load-balancing.
Even if the page is that - just a static page, the chance it's hosted by the content owner themselves is becoming smaller every day.
So frankly - yes. I believe today is the right time to start adding/replacing layers in the stack. After all - we've gone a long way in terms of software deployment tools so it's way more interchangeable.