Most of the "old school" hardware stores near me have been Ace Hardware or True Value stores.
Ace has been around a long time. According to Wikipedia, Ace hardware started in the 1920s with the first stores located in the Chicago area. It looks like they were close to a coast to coast operation by the 1960s.
Ace Hardware
True Value goes back to the late 1940s, and again looks like it was pretty wide spread by the 1960s.
True Value
ace, true value, and do-it best are all buyers cooperatives. The stores are independent, but the co-op gets volume discounts, stuff with the brand name on it, etc. The hardware store that is by my former office was one of the original Ace stores. It opened in 1880 or so. it changed ownership a few years ago, and they sold off a bunch of stuff they pulled from the basement, including a bunch of very cool hardware cabinets, solid oak, and with hand written labels. In German, because the founders were german...
People who didn't shop at Sears shopped somewhere else. Montgomery Ward sold pretty good tools, and had a pretty complete line into the early 80s, so did other department store chains. Hardware stores sold tools, of course, so did lumberyards, brickyards, supply houses for whatever trade. Automotive tools came from the autoparts store. The independent place my dad went to sold lots of blackhawk stuff, and had a lot of open stock wrenches, sockets, pliers, as well as pullers, and jacks, etc. When I was running a shop in the early 2000s, I bought a fair amount of shop tools, mostly specialty stuff, from one of the parts places we bought parts from. The outside sales guy there would tell me things like "the axle seal you're buying takes a 3 1/8 seal driver, you need one? What about the axle nut socket, that's 2 3/16?", and if I did, well, he got a sale. Hard to beat the tool showing up with the parts in an hour or so, and they had all sorts of stuff.

