The .410 has a couple of drawbacks to be considered: First, as said above, is the fewer number of shot in the pattern. That means "holes" through which a bird can fly.
The second derives from the first: In order to reduce the problem of holes in the pattern, most .410s are full-choke, which means a smaller pattern and thus a need for more precision and skill in pointing. If desired, a gunsmith can ream the barrel from full choke to, say, modified.
If a beginner doesn't understand about these factors, there's going to be a lot of disappointment at the numerous misses.
There are several things which can be done that are helpful. First is to try to make the fit of the gun as near-perfect as possible. For a nine-year-old who's just starting out, the stock most likely will need to be cut down to a proper length of pull.
Length of pull: With the arm bent ninety degrees and the hand in ready-to-shoot position, the butt should barely touch the bicep.
Proper fit: With a proper length of pull, the shotgun should be mounted with the eyes closed and a proper cheek weld is had. When the eyes are opened, the view should be right down the centerline of the barrel, just barely seeing the front sight.
If the front sight is not visible, the stock has too much drop at the comb or the shooter is mounting it too high on the shoulder. If the top of the barrel is visible, there is not enough drop at the comb or the shooter is mounting it too low on the shoulder.
If the line of view is not centered, the stock must be modified to correct that problem.
A competent gunsmith is your friend.
The 870? There are multitudes of them, new and used. They come in all variations from plain vanilla on up to very fancy. There are many add-on after-market options. They are very, very reliable whether new or used.