Just use applicator pads for tire products. I learned this the hard way -- I just broke down and ordered an entire applicator kit from autogeek. I have just about every common type of applicator on the market, and many of each.
Switching to an applicator designed for tires uses less of the product and makes the job so much easier an faster. (Oh and for your sake, not messy whatsoever) -- Zero gets on the ground.
But to the problem.. Yes the acid is a bit dangerous to work with, but just wear gloves, wear a mask if you have one.. goggles if you want -- And do it to it.
It WILL work =)
Have you tried household products:
Like
1) Bleach -- let is sit in the area for say, 30mins to an hour -- Or just let it sit and don't wipe it off.. Let it get removed or soak all the way in on its own?
2) Acetone (fingernail polish remover) -- Again let it sit, scrub if you want.. but this sounds like surface AND a medium depth stain.
3) Paint thinner -- Same process
4) Gasoline
None of those will hurt concrete.
I think the trick to this is going to be letting a product soak in like your tire shine has.. for quite some time.
I don't think scrubbing of any degree is going to do the job because its several mm's into the concrete.
Another option is paint..
The joking suggestion of doing the whole driveway with the tire shine isn't a bad idea. Be kinda neat to see water bead on your driveway rain after rain
Expensive if you have a big driveway hehe.. And that spot(s) will last longer than the application you do on the driveway hehe.