No Gak the police would not need to check everyone they see, it would just give them another reason to check suspicious characters for wants and warrants. Do the police stop every car they see crossing the center line on the road, no, but they can use that as a reason to stop a car they suspect of something.

Back shooting and ambush was not as prevalent in the old west as movies depict, that's not to say it won't happen but it happens some even now and would not happen more if we had open carry.