What am I doing wrong with the Database Tuning Advisor?
I wouldn't bother with DTA. It's a whole lot of work for results that are sketchy at best. Everyone that gets into index tuning eventually realizes it's both a science and an art, and there's a lot to it. You can use tools to get recommendations, but implementing those recommendations blindly almost always results in issues down the road.

Instead, I'd give BlitzIndex a try. It'll take advantage of the statistics SQL Server already maintains to determine where you can improve your indexing (and does so within seconds). It also provides very accessible documentation on how to use the results. Every result includes a link to the pertinent doc. This is how I maintain my indexes these days.