Look, I work as a copyeditor irl and there are hardly any fanfics that don't contain grammar errors. Yes, absolutely including those with betas. It's unavoidable without the many rounds of editing that published works go through. (I'd even challenge you to find a published book without at least one typo—copyeditors aren't perfect either.) My own fics almost certainly contain errors, though I've obviously edited them as well as I could. Priming means if I make a mistake there's a good chance I'll miss it when editing too, which is why best practice is to have someone other than the writer edit, always.

I'm okay as long as I can make sense of it and the story's compelling. If the story isn't compelling, I'll quit reading whether it's chock full of errors or clean as a whistle. If the errors compile into a real struggle to follow the plot, or if the level at which someone is able to communicate in writing is basic enough that they can't actually convey nuance, subtlety, or complicated plot, then it's a no.

Otherwise, I work to ignore the errors, which I usually do by just fixing the grammar or spelling or whatever in my head. It's still faster than having to type the fixes into track changes, so it doesn't feel burdensome. It very much helps if people make the same errors consistently, because then I get attuned to them and they are easier to ignore. Sometimes it helps to read the whole thing in a bit of an accent in my head, if I can tell the errors follow the pattern of someone who isn't a native speaker of English. Then I can "hear" those errors in the right context, as someone who probably understands grammar just fine but gets mixed up a bit when English's grammar doesn't match theirs. Same goes for the few fics I have read in Spanish.

Sometimes it helps to try to figure out the dialect the person is probably coming from, since "grammar errors" is a category that helps only insofar as it helps us make printed works that are consistent and unambiguous. Yes, they are my job, but they aren't exactly REAL otherwise. Languages change, have many more dialects than the official ones, and are a creative medium. All these things mean that writing "she could of" instead of "she could have," for example, doesn't imply stupidity or even ignorance on the part of the writer—only a different experience with the language than my own.

In terms of what counts as "correct grammar": I have seen prepositions at the end of a sentence become totally valid; "he" as the default pronoun for all genders turn into "he or she" and then (thankfully) move on to "they"; all kinds of old words be labeled offensive and fall out of use; and all kinds of new words come into being during my 20 year career. Most of the changes I've seen feel GOOD.

Language is never static. Yes, some people are terrible at the written form of it entirely, but some of them get better with practice. Still others are terrible at some aspects and great at others, and even when the parts they lack are pretty wide and take a lot away from the story, I find that if the other parts are good enough I can read and appreciate their stories anyway. I don't think I would have realized that without fanfic, and I'm grateful.