Not sure what little switch your talking about. Some gun's have a safty there, may be a selector lever. Most of the O/U's I've used have a tang safty with a selector lever built into it. The selector lever will allow you to choose which barrel you want to fire first. Sliding that lever forward takes off the safty and to one side or the other, which barrel will fire.
On my AyA's, SxS's, the little switch in the trigger guard is the selector lever. The safty is on the tang. I've a Browning O/U that has the selector lever/safty on the tang. I've never really gotton the hand of that set up. Mayby because I use SxS's most. The fastest and most reliable selector for me was doubble trigger's. When I was training dog's, I used a Charles Daly 12 ga SxS with double triggers and never wondered which barrel I was using. I wish my AyA's and my Browning had double triggers.
Probally hard to find but Ithica used to sell O/U's. I belivev they were called the model 300. They were made by Marouka (bad spelling) of Japan and were real nice gun's and cheap. They had a single non-selective trigger. You might find one of them around. Perhaps an old Marlin or a Pederson also. Pederson was made by Mossberg in their custom shop and were nice guns. I gunned for a guy at a firld trial years ago and used his. Liked it for on O/U.