What is it that you are attempting to argue?
Is anyone disputing that 1-2-1 is on average closer to mathematical reality?
I don't think so, though it is argued that this difference is overstated.
Are there angles of movement that highlight flaws in 1-1-1 ?
Absolutely, and I put them on the table from the start (45 degree angles or similar angles). In pure diagonal movement, 1-1-1 can extend range more than mathematically possible.
Are there angles of movement that highlight flaws in 1-2-1?
Absolutely. In steep angles where the number of squares moved diagonally are an even number. This can artificially limit range and causes one to be unable to reach a square that they should be able to mathematically.
If both have mathematical inaccuracy, is 1-2-1 less inaccurate than 1-1-1?
It's very possible, but it depends on how you move in a game. 1-2-1 is off a little less at small angles than 1-1-1 is off at 45 degree angles. If your movement is always in straight lines, 1-1-1 averaged being 1.0 square off on average in the examples I gave earlier. 1-2-1 averaged to being 0.5 off from the mathematical best square. A lot of real game movement isn't in straight lines. There is dodging around to avoid hazards, enemies, or other obstructions. The actual difference between the 2 is impossible to say as an absolute given so many possibilities.
What if a DM, really doesn't like 1-1-1? Will using 1-2-1 make the WotC knock on their door and cause their PHB to explode?
I highly doubt it. If it makes your group happier, then go for it.
What if a DM really wants it to be "realistic" and refuses to use 1-1-1 in their games, but some of the players in the game prefer 1-1-1 because they think it's easier and faster?
Do what you want, but it's generally a bad idea to override standard rules in favor of houserules that make the game slower, less fun, or more tedious for players when the players don't really want those houserules.
There's really not much more to it than this. Attacking 1-1-1 and trying to prove the mathematical accuracy is 1-2-1 is pointless. The answer is pretty clear, though I the degree to which it is more accurate might be debateable.
Likewise, I don't think there's a sincere argument that 1-2-1 is easier and more intuitive. Sure, it seems the pure diagonal corner case "tricks the eye" of people here not used to it, but they are also probably used to being shorted a square or two of movement also.
Really, this discussion has really broken down into us taking turns making statements like:
"I prefer 1-1-1 because I value a substantial increase in gameplay usability and speed is more important than a small and sometimes exagerated increase in mathematical inaccuracy. I also like Pepsi."
"I prefer 1-2-1 because I value a substantial increase in mathematical accuracy is more important than a small and sometimes exagerated increase gameplay usability and speed. I also like Coke."
Nobody can tell you that you should like something more than someone else, but it's pointless to take some kind stance of superiority regarding your preference. Both are reasonable choices, depending on what a group finds preferable, but there's no point in trying to bring up contrived scenarios to show that 1-1-1 has flaws, or to have a passionate dislike for it. It's just not rational.