I was considering a conversion from 32 rimfire to 22lr a while ago too. The gun in question was a Stevens Favorite. Naturally the take down design of the Favorite makes working on it easier than the Remington #4, but it should still be do-able. My project never materialised because I got into other winter projects and just let that one blow over.
I think you should go for it, if only because you can get the gun cheap or in trade. Once you have it in hand, you can try finding a box of ammo to put through it. Maybe you decide to leave it as is and lay in a couple hundred rounds for it. Then again, maybe you decide to get a barrel liner and make it into a 22lr. Whatever happens, step one is to secure the arm. Then you can decide in what form it goes forward unto the next generation.
The notion of 'temporary custodianship' of the artifact comes into play, for what it's worth. This item is already older than any of us, and may still last long after all of us are gone. Since it's already lost it's collector's value, the conversion to 22lr would make the gun more likely to be usable for longer. A hundred plus year old gun can be a wall hanger, or a rabbit killer depending on the owner's desire. If you fix it up in 32 rimfire, what's the likelyhood that 32rf will be available in fifty years? For that matter, 32 S&W revolver ammo is a bit less common than it used to be. The 22lr has the best likelyhood of being an available cartridge for generations. It is more likely to be exempted from ammo bans in the future than anything, and there are too many guns out there in 22lr to think that ammo manufacturers will stop offering it as long as they offer anything. I don't know how old you are or what year your specific #4 was made, but it's likely this rifle is old enough to have been your grandfather's or great grandfather's. I'm not saying that this IS a family heirloom in your family, I'm saying it's old enough that it could be. I'm also saying that a 22lr conversion would make it plausible that YOUR great grand children could still use the gun. Pure fantasy and speculation, but that is part of the fun of old guns to me.