I do. Most apps we write (government stuff) are CRUD apps that mostly handle data from the user, and then apply some logic to that data.
You can write custom logic in stored procs no problem. It's even a lot faster than doing it in java/php/ruby/.net or whatever because there are a lot less layers between you and the data, and the procedural language you use inside the the db is explicitly made and optimized for this purpose.
"The argument that you don't want to write any code that locks you to a database is some stunning lack of awareness, as you decide to lock yourself into the tsunami of unpredictability that is web frameworks to ward off the evil of being locked into a 20 year database product built on some pretty sound theoretical foundations."
Does Oracle or Sybase own Postgres? Nope.
There's nobody with a legal hand on your throat.
[0] https://cloud.google.com/appengine/articles/datastore/overvi...
I wouldn't use it a for a "normal" web app, for exactly the reasons you state.