I run it on an old laptop running FreeBSD and have my other machines use it as their DNS server.
See https://sockpuppet.org/blog/2015/01/15/against-dnssec/ for more.
BTW, there are nearly 13 million DNSSEC signed zones; the trend is certainly up [2].
Does DNSSEC mitigate the attack?
Yes and no, the server must implement strict DNSSEC check (i.e., refuse the responses that break the trust chain) to prevent the off-path attacks. However, since DNSSEC is still under development and servers need to accept such responses (i.e., only DNSSEC aware but not DNSSEC validate) when visiting a misconfigured domain.
Nobody is debating whether DNSSEC addresses this attack --- it does, for server-to-server connections. But DNSSEC is an extremely expensive way to stop this attack, and because there are cheaper ways that don't require collaboration between unrelated parties on the Internet, those are the mitigations --- ICMP policy changes, 0x20 encoding, DNS cookies, etc --- that will happen.
DNSCURVE was explicit about the idea of doing bottom-up protection of the whole DNS, so that servers would use it with other servers. You can, of course, do that with DoH as well, but it's not an "indicated use" yet.
So the short answer is: DoH & friends break this attack completely against home router forwarders (the first target in the paper), but not for major public resolvers like 8.8.8.8; they'll need to take some steps to break it themselves.
If you worry about DNS cache poisoning, sign your domains and validate the responses you get.