There's a whole design space of options here, one of the simpler ones being just checking via the registrar, using a protocol like RDAP. You could also use the DNS for this, but not require the entire zone to be encrypted (you'd be designing a new protocol, but that's what we have to do anyways, since DNSSEC is a dead letter).
replyYes. That would fall under the "some kind of similar PKI" category I mentioned, as you'd still need some kind of PKI to secure the RDAP connection. (Perhaps that PKI could be simpler than full DNSSEC, but it would have to solve a lot of the same problems.)
replyFor RDAP to work (for instance) you only need to bootstrap with secure connectivity to the registrars. Current DV challenges have a much harder problem: they have to work with every zone on the Internet. There's no real reason to accept that constraint!
replyWhen we think about "sound engineering" this is the level we need to start at, not "what features could we bolt on to core Internet infrastructure to define the problem away".
The people most avidly pushing DNSSEC for this problem aren't stupid. They're just smuggling in an additional premise: that the WebPKI itself is bad, and should be replaced with a DNS PKI. Seen through that lens, secure proof of domain ownership would be deeply problematic, as it would leave the DNS PKI with literally no reason to exist.