I prefer the .30-06 as it damages less meat, and you don't have to look for them.
Hmmm now the others can see whay I question your sanity
Meat damage has more to do with bullet selection and placement in my experience and if it was just the case of cartridge selection then following your flawed reasoning then perhaps you should be using the 30 Remington or 30-30 Winchester as they will or in theory should do less damage to a carcase than the 30-06
however it all comes down to bullet selection as I have personally seen the damage a 150 grain HP bullet from a factory Winchester 30-30 load on a Red Fox (Dog) at about 65 yards. That HP bullet at sedate 30-30 velocities made a hole you could put your fist in
and if fact I shot another dog Fox in almost the same spot from the same high seat but this time using a 303 using Winchester 180 SP amunition and it did a lot less damage with a heavier bullet at a higher velocity. The only thing it can be is bullet construction
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As for shot beasts running it seems to me that the ones that run are heart shot and know something is about and are are tense and ready to fly at the first sign of danger. Those beasts I have shot that were totally unaware on my presence drop on the spot or only take a few steps
the few that have run did not go far and the only beast I have shot with the 30-06 was already laying down after being hit by a car range was 2". I normally try for a high heart low lung shot as I found that they drop quicker but as I have not shot hundreds and only three that I recall being heart shots it's not exactly proof.
So far I have shot deer with in fair chase the road casulty does not count:-
6.5x55
243 Win
270 Win
7x57mm
30-30 Win
308 Win
The worst blood shot carcase was a Roe Doe of about 40lbs shot at about 90 yards with a Hornady 160 Grn RN bullet for the 6.5x55 the exit sholder was just jelly and un-usable
so as a result I have not used that load again on Roe Deer. My last deer shot was a Muntjac Buck of about 25lbs weight, a yearling and that was with a 130 grn Nosler Solid Base bullet from a 270 Win chambered rifle at about 80 yards the carcase made fine eating
these are small deer and yes both were fully grown.