I've actually yet to get an auger.

Chainsaw yes, but not an auger. (rng hates me there)
The pick is faster than using the axe for rock/iron etc. Mind you, without the ergonomic grip, and points in sexrex, yes, you'll run out of stamina.
(I tend to make a q3 or 4 one, and stay with it until steel)
The fire axe is faster than stone on wood, and of course, if you get jumped, you can whack the zeds with it then switch to something more lethal.
The shovel though, as I said, I'd have to check. I do agree the stam usage on the stone one is waay less than the iron one.
I DO run out of stam with the iron shovel, but not so much with the pick or axe. hmm.
I don't bother making steel tools below Q5, as making steel takes more resources, and of course, finding enough steel parts.
Plus, Q5 means 3 mod slots, vs 2. (so I can go ergo grip, bunkerbuster, iron breaker on the pick. Ergo grip is a must on all of them)
I may just spawn in an auger tomorrow, just to try the silly thing. Mind you, haven't found the desert yet, so saving the gas for the vehicles right now.
It's all on preferences though. What works for me, may not for you. (eg: if you can't find/buy/make an ergonomic grip, then yep, iron tools are gonna
hurt, and stone may work better for you)