I often wear a thong with my running shorts, my favorite is the soffe ranger shorts with the lining removed.  The lining to me isn't supportive enough and not comfortable for running, so the thong works a lot better.  My go-to is a joe sndyer regular thong (not bulge, or anything like that).  They offer great support, and I can match the thong color to the shorts.

On the few occasions I've run in only a thong on the beach its also been a regular joe snyder thong.  I almost always find something smaller for tanning / swimming, but have not found anything better for being active!

Thongzo, those g strings look like they will be pretty comfortable to run in! Like you said maybe for the shorter runs but I like how they don’t abruptly turn into a string. That’s the area I have issues with when I’ve run in a pouch type G string. It just doesn’t work for me and running. I’ll have to try those out!

Yes people run in just a thong. For many years I ran the beach in just a thong and a cap, three days a week, prior to injuring my ankle a couple of years ago.  Ran the beach in a thong for about fifteen to twenty years.

You asked about "backlash":  Now and again someone would say or shout something negative, but normally there was no hassle at all.  Normally the comments, waves and smiles were friendly and supporting. 
I never take a camera or phone to the beach, but over the years members of this board found photos on the web, and sent them to me asking if the picture were of me.  Some were, and some were not me. So now I have a few random pictures of me out running in a thong, or stretching after a run that were taken by others. I will post a few to my profile. 

Notably, unless I was posting for the picture, the photos people have found that were taken of me are all taken from the rear, as I disappear into the distance. Even I have difficulty with determining if it is me in some of the pictures. By the time they realized there was a man in a thong, and they got their camera out, I was past them and going away.

When I was still racing, (that's more than a dozen years ago), I ran in very short unlined shorts with a thong under the shorts. I did not like the liners so cut them out and just wore a thong. Some like the famous OnionSkin shorts were more than a bit transparent, so a person could tell I was wearing a thong under the shorts. I did not care for the built in liners on the IM OnionSkin shorts, so mostly cut them out. I often kept the matching shorts and appropriate thong of the right color together, on the same hanger. A blue thong under blue shorts made it look all one system. 

For running I don't like a thong that is triangular in front, i.e. has side seams going up the natural fold each side of the pubic hair. I prefer a narrow cut, closer to a teardrop, so there is no chafing at the top of the legs, on the part of the skin that is folding and unfolding with every step. Something like the old prevail-sport NorthStar or old skinz M14 cut in shape, but in soft fabric. Some of my racing shorts even had a built in thong. 

I got many complements along the way. I can't remember any negative comments about running shorts and thongs, even when the shorts were transparent and the thong very visible. It may help that most people don't criticize someone who is faster, and round here races start soon after dawn, so don't tend to attract the sort of person who makes public social comments about other people's fashion choices. The people at races are runners or families of runners. 

I do all the time in Pompano Beach. Pics to follow lol