I like the 444, but I think this is a case of the .45-70 being able to do anything the 444 can, but not visa-versa. Being able to use the same bullets in pistol and rifle is not that big an advantage. As mentioned, there is not really a bullet that works best, or is ideal in both the 444 and a .44 mag, so you'll be buying bullets especially for the rifle anyhow. I have found that the 265 Hornady will work well in both, as far as terminal performance, but that it does not shoot as accurately in my .44 mag pistols as I would like. It's designed for the 444, but will expand a little from a .44 mag. (really works well in a .44 mag rifle, but that's another story) As mentioned the pistol bullets really are too delicate for hunting in the 444, so at best you'd be using heavy rifle bullets in the pistol, which I guess is not a bad thing, but maybe not ideal, depending on how and what you use your pistol for. The 300 grain Hornady XTP's could be an exception, but I have never tried them.
Starting from scratch I'd go with the .45-70, it's "classic", big selection of cast bullets, VERY powerful hand loaded, but can be loaded down to any level you want. You can powder-puff it with 300 grain bullets, or go hog-wild with 500 grainers. The 350 grain bullets, loaded hot will give you as good a trajectory as the 444, and loaded to any power level are good all around kill anything bullets. I load the Hornady 350RN to about 1600-1700fps in my Marlin for all around deer and elk hunting. They can be pushed to 1900+fps pretty safely if you want more range/flatter trajectory, but I seem to shoot everything at about 100 yards.
For hunting with standard velocity .45-70 loads, the Speer 400 grain Flat Point really works good, really expands. Don't load it hot or it will blow up pretty good/bad on a deer at close range. Don't ask me how I know that.