Thanks for the help with the deletion of locked files. I was able to delete them by doing as you suggested.
After that and a reboot, restarting battle.net: Launching DI required 1min and 30sec. It’s much better than before, but pretty long nonetheless. Quitting the game, but leaving battlenet open and then restarting required “only” 35 sec. Quitting the game and battlenet both, then relaunching was 30 sec.
All these times are from the moment the button was clicked on the BN application to launch the game until the server selection screen was done loading and ready to use. Selecting the hero and starting the game requires another 46 sec. So after a crash, which occurs frequently, I’m now back in about a minute and a half. That’s an improvement, but still pretty slow.
Edit: I meant to note that I do have the game (and Battlenet) loaded on my phone, too. I don’t know if that is significant.