This approach does break some applications that pin specific certificate instead of relying on "any valid CA" model (e.g. Signal desktop) but that is seen as feature, not a bug, when it comes to enterprise.
Not on my phone.
I agree, however, that it should be possible to run your own _______. That's what the internet was meant to be.
eSNI and the encrypted TLS handshake proposal that was floated recently rely on fetching keys via DNS, so that's not applicable for DoH, and the handshake for a DoH client will probably be easy to distinguish from an HTTPS client.
It will be easy to block, if you want to.
If enough people block the published DoH resolver IPs, they'll see reduced availability on any hostname that is hosted on the same IPs; but if Cloudflare and Google put important content on the same IPs, it makes it harder to block.
This requires local client certificates to be installed.
1. https://en.wikipedia.org/wiki/HTTP_Strict_Transport_Security