"Should"? According to whom? Do we now change the dictionary as well? If we do, how do we make sense of the tens of millions of references to "class" that still exist and are a treasure trove of data in our past literature? Imagine your kid is doing a paper on the "new word" for middle class. When she Googles that new word - virtually nothing will come up. Her base of data will encompass 2023 forward. Not very useful.
Pretty soon, we'll have references like "Income Classified at Lower Levels than Average" and then someone will object to the word "average" as being applied to a human. It won't end if we start changing the language now every time someone imagines a word to be derogatory. Every word is probably derogatory in someone's mind.
I'm not a big fan of changing words that have been around for a long time with their meanings well understood - by anyone willing to understand rather than looking to be offended. That someone might now be offended by the word "class" is unfortunate, but not enough reason to come up with a new and improved, shiny, sanitized word that (in this case) primarily means economic status, denoted by income. We use "class" because we know it means "economic status denoted by income" and it's a heck of a lot easier to say "class" than "economic status denoted by income." And, we've been using it that way for 100 years or more.
I've grown up with the word "class" and have never given it a second thought about referring to anyone's behavior or societal value. That someone now does take the time to think about various connotations of the word, doesn't mean we have to change the word. That can get old pretty quickly. 'Cause then we'll soon be off to the next, new, more appropriate, acceptable word when someone new is offended by the old "new" word. Lower, middle and upper class have been around as long as I can recall and they've been used in news articles, economic studies and every-day speech as long as I can remember and I easily remember their use back to the 50s. I don't recall them ever being used as derogatory references. No word is perfect and every word can be misused. Having said that, lower class, middle class and upper class have come to be well understood whether someone can find a way to make them sound derogatory or not. Of course, since we all think differently about such things, YMMV.