If you want to use the WGRS you will probably have to change to a taller front too.
I have had 3 .22H's, and it is a grand old cartridge, but pricey as can be if you dont reload. I picked up a .22Mag and had it along with the Hornets, FWIW, they are now gone and it is still here. If I want/need more beans or range than it has I go to a bigger caliber, in my case 25-35, but a 223 would be dandy in a smallbore and you could afford Win. White Box and still handload it down to Hornet if you want less. A 22H Handi may be available with open sights and still the same issue of the front may not be tall enough, the 22M or .223 does not have them but the WGRS will screw on and only one D/T of the muzzle end will mount a ramp&blade.
My choice would be a 30-30 or 308 and load to whatever level of beans I needed, from 32-20 equiv. on up (though you wont get open sights on a .308). I already do this with a Win.94 carbine and have two loads, the 32-20 equiv. and the factory 150gr. that covers my bases. Why factory 150's?, because I use so few that the boxes I bought cheap on after hunting season sale years ago arent gone yet!
Most factory open sights are (and have been for some time) about as cheap as they can get away with. Fact is, they are just a point of reference and it doesnt really matter if they are cheap and simple as long as they dont break or just fall off. The older, inexpensive 22 single shots on the used gun rack have the same thing, cheap simple sights, and they have accounted for a lot of squirrels.