I think they specifically meant "should of" which is a colloquial form of "should've" in a number of places in the UK.
I went to school with a large number of people who would write "I should of done X instead of Y". In fact I'm pretty sure I made that "mistake" a number of times growing up.
Is there really a difference in how “should have” and “should’ve” are pronounced? There isn’t in any accent I’m familiar with.
I sure hope so, one's a contraction and the other is not...
And yes, there can be and often is a difference between saying "should've" and "should of".
(This response contains another funny modern colloquialism: 'a lot' gets used by both native and non-native English speakers where 'many' or 'greatly' is prescribed. Not that we need to be prescriptivists, of course.)
It also depends on the audience and medium, with "should've" being more appropriate for conversational/informal usage. It would be perfectly normal to say something like "he shouldn't've done that," but if I were writing a message, I'd at least expand the last contraction to "have."
I think there's a general perception that many of the common dialects of American English, especially in the South and West, are associated with being less educated. I am not sure where that comes from.
I'm a native English speaker, and my perception is that when someone speaks in a way where they don't use contractions, it seems verbose and stilted; I associate it with being scolded or disciplined, or when someone is speaking sternly to make a point (or out of anger). E.g.: "You don't know where you're going, you should've taken a left" - informative/neutral "You do not know where you are going, you should have taken a left" - critical/scolding
Omitting contractions can result in speech that sounds strange and unnatural in general: "Shouldn't we go?" -> "Should not we go?" "Aren't you coming?" -> "Are not you coming?" "We didn't, but we should've." -> "We did not, but we should have."