I have a Browning BL-22 as well as three different variants of the 9422
9422XTR and 9422 Trapper
9422 Legacy
All of them are very well made and are heirloom worthy rifles.
I can’t say the same about the Henry rifles. The design is a slightly modified version of the Ithaca Model 72, complete with zinc alloy receiver and cosmetic sheet metal receiver cover.
Now…like the Ithaca 72 they are not bad rifles and the assuming the zinc alloy is similarly free from impurities it should age pretty well without cracking or warping. But they are not heirloom worthy rifles.
Subjectively the Browning BL-22 and Winchester 9422s feel much better in the hand and when operating them the higher quality speaks for itself. They are also accurate for the type and 3” groups at 100 yards are the norm.
The BL-22 lacks a proper metal tang so tang sights are not an option, but you can still mount receiver sights and they accept a scope just as well as a 9422 or other short receiver .22 LR lever gun, which is to say, the ring placement sometimes ends up being unconventional. Unlike a Henry, the scope or receiver sight is solidly mounted on the receiver with no receiver cover involved.
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Some of it comes down to your kids. If they have are prone to pride of ownership and take care of their stuff, you want the BL-22. They’ll be proud of it and keep it.
If your kids are more prone to abusing their stuff, or guns and heirloom considerations are just not their thing, they will probably turn it into money the first chance they get. In that case get them the Henry. It won’t bring as much at a pawn or gun shop and the reduced incentive to sell it might mean they keep it longer.