, I don't know where you got the idea that your rifle won't "stabilize" a LIGHTER bullet. It's actually the other way around. Twist rate is figured on weight AND length of a bullet. The longer, heavier bullets need a faster twist rate to stabilize them. So a longer heavier bullet will be stabilized LESS.
The 30-06 is figured around a 180 grain bullet. Also, that means a jacketed LEAD CORE bullet. It will work for heavier bullets, up to 220 grains.
All-copper bullets like the Barnes, Hornady GMX, and nosler e-tip are longer for any given weight. Length matters more for whether a bullet will be stabilized or not.
The GMX and E-tip are actually made of guilding metal, you know the stuff they make jackets out of? Barnes are made of pure copper.
If you have a normal 1-10 twist in your '06, then the lighter bullet could be a bit OVER stabilized. That CAN lead to problems with getting a---lets say 150 grain lead core bullet to shoot small groups.
As others have noted, using a lighter all copper bullet as opposed to a heavier lead core bullet will give you all the power and expansion of the heavier bullet. That's because all copper bullets don't shed weight, the weight retention is usually 100% That results in deep penetration.