I just wanted to say thank you for everyone's clues. Added up, it worked.
For anyone having issues, here's my full report on what worked for me:
Samsung Galaxy S3 T-Mobile 4.1.1 ota update as of 11/15/2012
Bricked device 11/20/2012 attempting root via the toolkit after multiple failures using Odin.
Recovered device using full image (755MB) that I got from samsung-updates.com via Odin.
Successfully rooted using Odin with the Sprint labeled .tar.md5
Installed Busybox and SuperUser
Tried 9 wifi tether apps with no success including the same but UNMODIFIED version of the one in the Original Post.
Installed the OP apk and was prompted about 15 times to allow root access.
Successfully connected wifi tether once but could not duplicate.
Read thread and installed SuperSU (I should have done this before any of this anyway)
Reinstalled the modified-wifi-tether apk from the OP
Successfully tethered now. I call this "perm-tether", as I can duplicate it now.
Settings are:
Switch device to Samsung Galaxy s3
That's it.
SuperSU really worked the magic here. I'm not sure if the timing was off before using single su's but superSU approves all the requests and it links right up every time now.
I also have an additional piece to add relative to our phones. I was able to get flash working with a work-around. Fortunately you won't have to do that. I have extracted the working flash apk and verified it on 3 sgs3s and 1 note2. The flash apk i have, will get your flash working again. I will upload it as soon as I can if anyone needs it.
Details of the workaround I used for flash:
After trying literally 9-10 different apks for flash that claimed they'd fix/restore my flash, I gave up.
I decided to go through my old downloads via Play and see if there's any games/apps I didn't realize I miss.
I saw Flash in there from when I had my g2 and I downloaded it via Play. I clicked Update and sure enough, it took.
I immediately got apkextractor so I wouldn't lose the ability to update it (if Google pulls the update files from Play, which i would expect)
After extracting the apk, I sent it out to 8 people. 4 responded that it worked for them. 4 didn't respond at all so I assume it worked for them also.
If you're missing flash but had previously downloaded it, go into your Play downloads under "ALL" and scroll down and update it, enjoy flash again.
Again, if anyone needs that apk let me know and I will get it uploaded.
Thanks again for everyone's input, I hope my over-view helps someone.
PS: I think it's absolute SHIT that any carrier charges for tethering as a service. It's a 'native' feature of the phone's capabilities, not the phone carrier or service. Furthermore, carriers (most) limit the amount of high speed data we are allowed anyway and that should be reason enough for them to allow us to waste it in whatever way we want. Whether that be tethering our entire home to it or downloading apps and loading ads. I don't consider most sgs3 owners the typical "cheap" crowd, and that argument doesn't go far with me. Those of us into modding our phones at our own risk should be granted the "rewards" that come from the risks we take. If we root to tether, let us tether. The carriers are never going to be affected by it because they limit us anyway. Just my opinion on the matter anyway.
Cliffs: get SuperSU before installing this modified version of wifi-tether to increase your chances of success.