I've done ALLOT more walking around looking for grouse than actually finding them...so that doesn't make me an authority by any stretch of the imagination. With that said, I've popped maybe enough of them to form an opinion on what works best for me. As others have pointed-out, there is no such thing as a perfect grouse gun, but you can get close by making sure the gun (and choke) is well suited to the areas you hunt. Personally, I think 20ga is a great combination of power/reach/lesser meat damage...and ammo is affordable for both practice and hunting. I've been curious about 28ga, but probably wouldn't spend that much $ per box of shells practicing on clays. lol Anyhow, when I've tried TOO hard to match the gun to the areas I hunt...it hasn't always worked-out.
In my main spot, it's thick bush/sitting birds (hopefully) at not too great a distance. After shooting a couple like that with my 12, I switched to 20ga, MOD, 28" barrel
Took that gun out, same scenario, too much meat damage for direct hits at close-ish range. Bought a spare barrel, had it cut to 22", CYL bore. Shot 1 bird, maybe 2 pellets in the breast total~thought that was fantastic. Every outing after? All birds way too far off, and I even missed a relatively close wing shot. (blame that on it being a shorty)
So, after that...just decided to focus less on the gun, more on the hunting. What I'd buy if I were the OP? Probably a 20ga pump. New, sub $500=Remington 870 Express or Mossberg 500. A single shot for close to half that price~maybe look at one of the Stevens (=Savage) 301, single-shot 20ga. Like a modern Cooey 84, only comes threaded for choke tubes. Suggesting mostly due to price/availability (saw one online @ a Canadian retailer for $245 new) but ALSO because I've only ever had a good chance at a "double" twice in about 10 years. So, 10 years of lugging around heavier guns IN CASE I have need for a quick follow up...and have only needed it 2 times. OF those 2 times, I only scored a double 1 time. A single-shot would not have changed the outcome of most of my grouse outings.
Used? Guns like the Ithaca 37, Remington 870 Wingmaster***, Mossberg 500 in 20ga. or even 16ga. if you don't mind spending about $3-$4 more per box of shells. All tried/true/trusted. I've owned all 3, and the nicest to carry is the Ithaca. The easiest to work on is the 870. The least expensive is the Mossberg.

Does that help? What am I most likely to grab walking out the door grouse hunting? Ithaca. For single shots, a Cooey 84 would be the lowest cost, but you see mostly fixed/full choke guns out there. (mine is a 20 MOD) If I was buying today, I'd probably try the Stevens 301 for the choke-tube benefits.
***The OP mentioned price, you can find 20ga "large frame" Wingmasters for less money than the LWs, problem is...not everyone selling their large frame Wingmasters reveals that they are LF guns...and are looking for LW money. Maybe some don't know...not sure.