You're absolutely on the right track with the .30-06. As far as a versatile gun that will handle almost any game (with the proper ammo), you can hardly do any better.
The 30-30 and 44 are both marginal guns for your need, and IMO, they are relics of the past. The were good--even great--back in their day, but time moves on. The main advantage to the 30-30 was that it's lightweight and easy to carry. With the advent of light bolt action guns, that advantage has been nullified.
The 30-30 and the .44 have to be considered to be 100 yard guns for all practical purposes. Now we'll pause while the 30-30 fans pile on with tales of their great great great grandpappy's 500 yard shot right through the eyeball of a running elk. OK, we're back in reality land.
The .30-06 can place accurate and killing shots cleanly and efficiently even beyond 300 yards. Even at closer ranges, it is going to be superior hunting rifle in almost every situation.
Grouse