I crewed tanks in the mid -70s and agree with TSMGuy that the M3 was considered junk. We had both M3's and M3A1's. A Sten has more class (though not more clout). Having carried one in the military, I have no idea why they command such high prices in the collector market.
Sure, they're semi-rare, but so are my old toothbrushes, and nobody's offering $15k for those.
We used to adjust the rate of fire by how much LSA we put in them. I used to carry two bottles in my ammo pouch (we were issued M16 ammo pouches for our M3's and 1911's. Go figure...).
We fired them only for familiarity. I think the requirement was annually. We'd pour in the LSA, and let 'er rip. Normal rate of fire was around 400 rpm, and the oily guns seemed to go more like 600 (I wanted to say 700, but decided to be conservative).
Smoke would just be rolling off them (the oily ones), and we made no friends out of the next shooter on our right, whose back would get sprayed with LSA.
Messy, but fun. And we didn't even have to buy ammo!
Yes, they're reliable and cheap, and made good sense at the time (WWII). But collecting isn't about making sense (I guess that answers why they're so expensive!). Compared to a Thompson, including an M1, the M3 is almost Soviet in design: crude, cheap, not much to look at, but did an OK job for what we were willing to spend on them.
Those aren't the qualities I look for in a collectible firearm. Those are more like the qualities I'd look for in a date.
My $.02 would be Thompson first choice, MP40, then do some bargaining to see how much he'll pay YOU to take that M3 off his hands.
One of my tank commanders used a grease gun in Vietnam and really liked it. He used it to shoot at something other than paper (which is all we did -- kind of like the Texas Air National Guard at the same time). He'd know better than I how good a weapon they were in combat.
Sorry if I've offended anyone who owns grease guns; it ain't about owners, but about the official replacement for the Thompson. Yeah, it replaced the Tommy in the TO&E, but not the heart. (Put both on a table at a gun show and see where the crowd goes first...)