It can be OK, can’t it? Or have I just been here too long?
I usually explain it as “because part 2.” “Don’t bother trying to go to that restaurant: the food’s overpriced. Besides, it’s so popular you’ll never get a table anyway.” It’s for cause & effect patterns, when you’re introducing an additional cause.
To me that’s the meaning used in the examples provided in that Cambridge link above, although the link doesn’t go into that. They’ve got “It’s quite cold, and, besides”, which grammatically is fine to me, but I don’t see why “It’s quite cold. Besides …” isn’t equally correct.
As a linking adverb, we usually put a comma before and after besides in writing:
I don’t think going for a walk is a good idea. It’s quite cold, and, besides, it’s getting late and we don’t want to be out in the dark.
EDIT: note how the confusion with “in addition” makes some sense. In the sentences above, “in addition” would be OK as a substitute for “besides”. But what happens is students will write things like “I went to France and Germany. Besides, I went to Italy”, which doesn’t sound right to me in most contexts.
I have definitely had more trouble finding useful grammar links in recent years. Oh boy there are a lot of badly written and incorrect sites out there. My operating assumption has been it’s out of China, but I don’t know if that’s true.
I think one aspect of the overuse of ‘besides’ is from Chinese writing, it’s a translation of “並且”. I’ve translated a lot of Chinese paragraphs that are not particularly well-written, often it’s one long run-on sentence, kind of a list of points the author wants to make but without organizing them or arranging them in a sensible sequence. When translating such passages I find myself looking for places to break the list and put in proper transitions so that a point is being made instead of it just being a list. And whether I’m translating or just editing someone’s English, when I see ‘besides’ I know that often what they mean is ‘in addition’.
I got the impression that “quality” “advanced” Chinese was when a single sentence continued on for pages and pages. That’s why “advanced” English texts at “good” cram schools usually involve something like dolphins, which are not fish but are actually mammals, like you and me, even though they swim in the water, but they are still mammals that live in groups called pods and a lot of people think they’re really smart because they can make music, but not instruments like humans but rather they make squeaking sounds through their blowholes. Or something like that.
One time, when I taught the whole “your composition is like a hamburger” analogy to junior high kids, my co-teacher needed to butt in to emphasize a point about the bottom bun. She said “Unlike in Chinese, you don’t add any extra information in this part. So in English, you don’t add further details in your conclusion”. So it’s OK to just throw in a quick “BTW” in your concluding sentence in Chinese?
Not me. I’m much leggier and much hairier than the cut-price callgirl in that pic. Besides, I only ever wear micro-minis when on the job: I like to give clients a 若隱若現 peekaboo glimpse of my imported “salty penis”