YSK it's either "should have" or "should've". "Should of" is incorrect
Why YSK?
The first person who typed "should of" probably heard of it in real life that was meant to be "should've", they typed "should of" online and readers thought that it's grammatically correct to say "should of" which is in fact wrong and it became widespread throughout the years on Reddit.
I hope something could start to change.
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...the reason "in some dialects of English native speakers really do say 'should of' etc" is phonetics. Kids hear "should've" and repeat it phonetically, before learning the actual words or their meaning. Combine that with the awful state of education and literacy in the USA (and other countries etc) and voila, you've got some armchair internet expert justifying it with some big words trying a weeeee bit too hard to make it work.
Then you've got teachers who still gaf and know their shit who will correct this before middle/high school, and no, last I checked it was never added to the dictionary or considered correct. Language of course is living and ever changing, but the line must be drawn somewhere lest we devolve into shouting and grunts like neanderthals
This is the failure of "no child left behind"; it seems that's all it did !
What the author of the first link claims (and the second link explains in a more accessible way), is that it's not just that for everyone. Like some native speakers really do say "of" sometimes, even when it's stressed and doesn't sound like "'ve" at all. So for them it wouldn't just be a spelling mistake, but a different grammatical construction.
Some dictionaries (e.g. Merriam-Webster) actually do list "of" as an alternate spelling of "have" (not in the sense of a genuinely different grammatical construction though).
Obviously it's not considered standard by anyone, which is also why teachers call it incorrect, who (should) teach the standard dialects.
Language changes whether you and I like it or not, and efforts to stop that from happening are generally unsuccessful. You can also rest assured that a simple change in what is considered correct grammar or spelling (which, as far as I know, nobody has been suggesting in this case so far, but it seems like that would be the "worst-case" scenario from your perspective) would not lead to us or language "devolving". Also, while we don't know anything precise about how Neanderthals spoke, most likely they sounded more or less like us and did not communicate by "shouting and grunts".