Several years ago at trust school I roomed with a fellow whose bank sent statements in plain brown envelopes without even a return address, but he was from the Bahamas...
Just a personal opinion, but I don't think mail thieves sort the mail when they steal it - they just take it all. Removing your bank name and logo might be a subtle indication that you strive for customer privacy. However, as you note, many people, myself included, sort the mail with a shredder between their knees. If it doesn't look like something I want, zzzzzpppptttt, unopened.
In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.