I rinse off the heart and liver before eating, because there are usually some blood clots encountered when trimming. I wouldn't call it washing. I don't want to eat a nasty blood clot.
I salt brine most deer meat in water and ice for 3 days, changing out water. Drain the water and let the surface dry off before vacuuming and freezing. This is fuddlore passed down from my FIL. I think there is probably a 3 day aging/tenderization process, prior to freezing, that he was right about by accident. Wet brining can remove hair, leaves, dirt left on the muscle. If a deer happened to be gutshot, I would rather brine it to reduce the quantity of spoilage.
The meat is not a sponge. It can only get so saturated with water. It cannot get "waterlogged", it will take on about 5-6% water weight. There cannot be any more "flavor" exchanged in the solution than the absorption of water. The salt will not significantly permeate the meat. If you drop a whole muscle roast in a salt solution with food coloring then the coloring will only penetrate the outer surface within an inch of depth. Same with a dry brine, dry rub, or a smoke ring on a roast. Unless you use an injection, a solution cannot penetrate very deep in whole muscle.
Sometimes I take a deer sirloin or round that has been wet brined in a cooler for 3 days out of my freezer and dry brine it in Mortons Tender Quick (quick cure based on weight) for 5-10 days. Put a pickling spice dry rub on it, smoke it rare and make deerstrami. It's not overly salty and its way better than some bullshit corned beef packaged in cryovac at the supermarket.
The people saying brining, or soaking, causes significant flavor loss are just as full of fuddlore as people that soak and brine. I think some people see a little bit of surface discoloration on the outside of a whole muscle and think, oh it's ruined and it's probably lost all this flavor. You can try this yourself and see that it's not really the case.
This is a repeatable experiment. One can salt water brine their meat, dry brine, dry rub, or cure meat.
Wet brining or a light rinsing isn't going to ruin meat. If you can't put a sear on it then your fire isn't hot enough.