I have a very old rock maple chair that was stained at the mfr when it was made, probably circa 1910-30. I have done everything short of an atomic bomb to get this stain out of it. I have decided to "go with the flow" and get as much stain as I can off and live with a sort of mottled look (which sort of has a "grain" look to it, like birdseye maple) but I was wondering about lightening certain areas with a mild mix of muriatic acid just to get a more even tone all over. What do you think?

While some professional woodworkers use it sparingly such as in a vapor or light wash, you need to know that it is basically Hydrochloric acid (in water) and that it will sink in and 'eat' the wood. It attacks cellulose, and wood is cellulose.
I can't imagine why you want an old chair to be unstained. Many late 1800s and early 1900s wood items were factory made, the beginning of mass production, and were stained BECAUSE the woods didn't match grain and color. And to try to strip that many little corners and connections is very difficult.

I meat oxalic acid, sorry.

Oxalic acid probably won't work if used in a 'mild mix' but sure, give it a try. I don't know if you got all the varnish off or not. That might be why it looks mottled.