1) How would have a “chumra” that is directly against certain mitzvos, specifically korbonos. It is not the same as not blowing shofar or taking the arba minim on Shabbos. There you are not saying that there is something wrong with the etzem mitzvah.
2) A chumra is to protect against being oiver an aveitah or being mevatel a mitzvah, what aveirah or mitzvah are you supposedly protecting your self from by not eating meat? There is no issur of tzar baalei chayim when you are shechting. There is no reason to even go into the reason of me’od me’od.
If the person said he is not eating meat because he is concerned about kkashrus, and therefor he is machmir not to eat any meat, that could be a legitimate application of chumra, but that was not the focus of such an alleged chumra, so it is a bogus application of the concept of chumra.