Depending upon bullet shape, the standard 1 in 10 twist WILL generally stabilize the 220 grainers - there are factory loads available with 220s even. These aren't particularly long-for-weight bullets, but are shaped more or less like "solids". My understanding is that they will do, penetration-wise, in UN-bonded form, the same thing a quality modern BONDED 180 will do, and do it cheaper, but at the expense of some trajectory.
But as for 240s & 250s, my understanding is that a faster twist is needed - 1 in 9 or better, depending upon bullet shape. These are more of a specialty bullet which will shine in something like a .300/.221 whisper, with a high twist rate, shot at subsonic velocities, where you need bullet mass and BC to make up for the puny velocity, for both external and terminal performance.
I have actually considered buying a *second* .30-06 rifle, in order to have this one long-barreled one specializing in 220s for stuff like hunting grizzleys or hunting other game in heavy grizz country (to maximize power/energy), and one short-barreled "standard" one for general use such as elk, with 180s. I like to have ONE load that I settle upon for each rifle, not change loads for different uses. A lot easier to change rifles than re-zero your optics for different loads out of the same rifle. But ultimately it made more sense to step up to .35 whelen / 9.3x62 range for the serious canada/alaska rifle instead of two specialized .30-06s, though the idea had some appeal due to ammo cheapness/availability. Oh, 2000-2200 fps is *plenty* o' velocity to kill stuff dead - the unbonded lead 220s will flatten readily at such "low-ish" speeds. 200 fps is more than I'd stand in front of for a 220 gr high-SD bullet.
