I definitely agree on going to .223 or .22-250.
A .223 comes standard with a 12” and many are offered with faster twist. A 12” twist will stabilize the 60gr Part, 64gr BSB and 64gr Winchester PowerPoint and well as some monos. An 8” and maybe a 9” should stabilize the 70gr AB.
As mentioned above, in the case of the .22-250 you’re going to have to be careful with the twist. It comes standard with a 14” twist which will not stabilize any of those bullets. I’ve tried. The .22-250 made a name for itself killing deer, but we don’t have the bullets they were using back then. The original 60gr Part would stabilize in a 14” twist. The new one is longer and will not. Sisk made a 49gr bullet that was mostly copper with lead tip and could be run up to almost 4000fps in a .22-250. It was like shooting a mono with an exploding tip. Nosler made the Zipedo in 50gr and 55gr, which was like today’s BSB but not bonded. It was similar to the Sisk, but with more lead for expansion and less copper for penetration. On a side note, th original Solid Base was basically today’s BT Varmint except tipped with a lead soft point. With a 14” twist you’d be limited to a light mono or neck/head shots. There are some tighter twist guns from the factory, but they’re not nearly as common as tighter twisted .223’s, so if you’re looking used, it might be hard to find. You can always buy new, or of course have a gun rebarreled in any twist you want. Factory ammo in the hunting bullets may also be difficult to find. The .22-250 offers substantial advantage over the .223 if you get one with an appropriate twist and enjoy handloading. Otherwise, the .223 is a fine deer cartridge for youth hunters.
My wife killed her first deer with a .222Rem using a 55gr Lapua soft point, and I have killed many hogs with the same rig. It gave 18”-24” of penetration and a substantial wound channel. That’s not enough penetration in my book for mule deer, but for a central Texas whitetail it’s was plenty. The same bullet from a .223AI, used on a hog, penetrated about 4” creating a softball sized and shaped lump of goo. You’ll have to have a far tougher bullet to be effective at .22-250 velocities.