I’ve taken a few youngsters out, so I’ve answered this questions myself over the last few years.
Of those choices, I think that .410 is too small right of the bat. It could certainly be effective, but I’ve learned that “room for error” is nice when it comes to beginners.
I’m a big fan of the 20 gauge for deer hunting, for any size, gender, and age. With the loads that are available nowadays, there is plenty of options and plenty of killing power.
Muzzleloader would be nice for being able to control powder content, but of course there is no follow up. Would certainly teach the youngster the importance of the first shot.
Best option in my opinion? Scoped .243 rifle. This depends on the area you hunt, but it is really hard to beat a small, light, .243 caliber rifle for deer.
When I turned 12 I remember my father buying me a 12 gauge, as my first and only shotgun I ever shot back then. I had to rest it on logs to shoot deer, but it worked. I also think this is a quick way to encourage poor shooting form and wounded animals due to flinch.
20 gauge wins the vote for shotgun zones, .243 for rifle.
Hope that helps.