Everything already works everywhere for everyone with regular DNS. The only thing that is broken is firewalls. Specifically, nobody wants to force network administrators to fix them, so instead they're trying to work around them. Not only is this unnecessary, it's stupid, and just a cheap workaround for something which should have been solved by the industry 18 years ago, but is now left to be wrangled by 3 companies with leverage over users.
So I had a DNS server that used UDP. Now I need a DNS server and an HTTPS server that uses TCP. Now I'm encapsulating DNS in HTTPS. So now the packets are bigger and there's extra latency, increasing traffic and slowing things down. (These kind of "fixes" must be common, because I seem to notice how technology is constantly getting slower and crappier rather than the opposite)
Then there's security concerns, because you're running twice the software with greater complexity, yet it's subject to the same attack methods as everything else, meaning one exploit could impact DNS just as much as every other HTTPS service. Among these are attacks on the PKI system, which is fragile to say the least - an expired or revoked cert kills your DNS, and any CA can generate a cert (which apparently isn't hard to do using a BGP exploit) which can be used to circumvent the encryption. Since you need DNS to do anything on the net, it will be trivial to simply kill all DNS-over-HTTPS queries going over a network and force people to use a downgraded, insecure version of DNS. And you've got to handle encryption in a way that is normally for streams rather than datagrams - unnecessary for data which only really needs authenticity and integrity assurances (which is, again, already provided by DNSSEC)
Then there's reliability. You're tunneling DNS over a connection to one server. We already know this is slower, and we already know DNS is one of the factors that slows down page connections. So you had better hope that the one HTTPS endpoint you have a persistent pipelined connection to doesn't go down, or wait and wait for what you hope is an independent 3rd party's service that is also in your list of DNS-over-HTTPS resolvers to pick up the slack.
As a DNS admin, I had software that communicated with DNS servers. But now I have to rewrite everything to use DNS-over-HTTPS, because if a query fails in DNS-over-HTTPS, you have to troubleshoot the actual connection and not what you think is in DNS, or what you think DNS should be sending back to the client. So now I have to reinvent every DNS tool. And of course, for the enterprise DNS solutions that can't just magically sprout a new protocol extension, I'll need some sort of middleware software to map the DNS-over-HTTPS queries to the real DNS server.
All of this because someone wanted "privacy" of their DNS queries, and was too lazy to allow UDP packets greater than 512 bytes over their network.