I'm personally still unsure of how to feel about it. I see some possible benefits, and also some privacy concerns. With all that in mind, I decided to make a JavaScript library a few months ago to make it easy to do DNS lookups from within web apps. It's called DoHjs (super creative name, I know). You can check it out on Github - https://github.com/byu-imaal/dohjs. There's also a page you can try sending DNS queries directly from the browser from called https://dohjs.org. It's far from production ready, but it's got some really cool features right now with more coming soon!
I'd love to hear some thoughts/feedback - could be about DoHjs, websites doing DNS, DoH in general. I suspect many will be interested to use DoHjs, perhaps for different purposes than bypassing people's default DNS resolvers. Feel free to comment here what you think, or open up an issue on Github if that makes more sense.