I would want somthing that could perform at 350 for elk when in reality it will be around 250-280 yards.
I have already used a .30-30 and a 7 mm rem mag to take deer at over 220 yards. All it took was time at the range.
Again you have a 7mm Rem Mag, that is all you need to take an elk with at 300+ yards. You are already comfortable with that rilfe so that is the one I would use to hunt elk with. If you are just looking for a different rifle, I'd go with a non-magnum rifle that fills a gap between your .30-30 and 7mm Rem Mag.
FWIW the .30-06 with 180 grain bullet will have almost the same energy at 350 yards as your 7mm with 160 grains. So the .30-06 isn't lacking any power when it comes to killing elk. The .30-06 just takes a little longer to get there, but the elk won't know the difference.
I like the shooting sticks for hunting, if you are stalking they take less time to set up than bi-pods. If you are setting and waiting in a lane then bipods work just fine. My Stoney Point shooting sticks are much lighter than my Harris Bi-pods, and light has advantages when humping the hills for elk.