For me, much depends on the area I'm hunting. If it's full of trash, I'll eliminate most of it out, BUT....I do this because I have trouble getting up and down, and by the time I'd dug all those pull tabs, or pieces of foil, I couldn't dig a gold ring anyway. I guess you can say I "cherry-pick", but it's because I need to make the most from each time I get down on one knee. Honestly, this is a large reason I don't hunt parks, or trashy areas much, just not worth it to me. I'd rather spend more time researching cleaner older homesites, schools, churches, etc.
On sensitivity, I seldom run over about 4 bars. Generally, old sites I hunt are full of iron, nails, etc, so I'll eliminate it out as well. I'll start leaving nickles in....but if all I dig is trash, I'll notch them out too. Again, my style is probably different than most because of injuries. Have I passed-up on some good targets? Maybe, probably, but sometimes I return (if I'd made some decent finds on the 1st hunt) and hunt it a 2nd or 3rd time after I've rested-up and can dedicate my energy to digging the more "iffy" signals.
Smitty