I used a single-shot .410 to hunt grouse in northern Idaho years ago. It did the job, within 25 yards. No problem. I used No. 6 shot because I was hunting in heavy cover.
I gave it up, though, when I purchased a Remington Model 870 Youth Model pump shotgun in 20 gauge. It's a magnificent shotgun for hunting small game in the hills, where a lot of walking is required. The handiness and light weight are appreciated. And with a slug or buckshot, you can still take deer or defend yourself against predators -- four and two-legged.
Mine has a 21-inch barrel with interchangeable chokes. It's a Plain Jane model with hardwood stock and Parkerized finish. No fancy wood or finish to agonize over if it gets a ding or scratch.
I went to the 20 gauge because I occasionally had to make shots longer than 25 yards, as grouse flew down the length of a logging road. The .410 just didn't have enough pellets to reliably do the job.
I added a black nylon sling. Makes it handy to sling over your shoulder if you need both hands for climbing up a hillside.
But I well understand your attraction to the .410 shotgun. They're cute and scream FUN! I gave my single-shot .410 to my brother. Later, I was given an old Marlin 410 lever-action shotgun made about 1929. I bagged two grouse with it some years ago.
The .410 will do the job on small game, within 25 yards. Beyond that, the pattern gets thin by virtue of having relatively few pellets compared to other gauges.
I've always liked the looks of a .410 double. You may wish to consider this, instead of a single shot.