Were real, Western-style horse-drawn stagecoaches still used at all in the 1920s?

No, for that the hotels sent a particular kind of wagon, often with a rather fancy appearance such as wood paneling around the cargo area . . .

what came to be called a station wagon.

Within a city, you would take an omnibus.

Now that the question has been more or less answered, I want to share a “modern” stagecoach story…

I was camping in the desert outside of Barstow back in the mid-90s, it was mid-week and I had the whole campground to myself. From my campsite up at the top of the loop I could see the whole campground and had an expansive view out over the surrounding desert. Right around sunset I spotted headlights coming up the access road and as the vehicle approached, it was clear that it wasn’t a normal car. I had to do a double take when it drove by…it was a motorized stagecoach!

I wandered over to take a look and found out it was a local body shop owner and his pals out for an evening campfire dinner. Greg Parker was the guy’s name, and it turned out he was pretty famous around the area for building Mad Max-style vehicles, some of which are still visible on the side of old Highway 58 outside of town.