In this context, you're almost right in that she's saying they don't care for them anymore. But it seems to me that what she's getting at is, they aren't even pretending to care, hence sending someone who is so obviously unqualified to do a job.
Brace yourself now, because things are about to get literal:
"You see, they're just pissing on us without even giving us the courtesy of calling it rain."
Imagine that you're walking down the street, minding your own business, when all of the sudden you find yourself sprinkled upon by some wet substance that smells like pee. After hustling out of the way (how gross!), you look up and see an old man peeing off of a balcony. He's totally oblivious to the fact that he just peed on you and finishes up, and then after a few moments looks down at you and realizes what he'd just done. He quickly looks around to see if there's something else that he could place the blame on, and eventually settles on blaming the weather. "Well now, it looks like we've got a bit of rain today..." he calls down, and then hurriedly rushes out of sight, realizing that he'd just done something totally wrong.
Although lying about it wasn't courteous at all (calling it so would be a grand example of sarcasm), let us contrast it with the following situation:
The next day, everything above happens again down a different street and with a different old man. But this time, after he finishes up, he looks down at you. You glare at him, expecting maybe an explanation or an apology. Instead, he locks eyes with you, grunts, and shakes a little bit more out before sitting down and reading his newspaper. You see, he's just pissing on you without even giving you the courtesy of calling it rain. In this context, hopefully the meaning is more clear.
A more common use of "give/show sb. the courtesy" would be:
You could have shown us the courtesy of letting us try to fix it before giving us a bad review on Yelp.
This could be a business's reply to a customer's negative review on yelp, if the business owner felt that it was an unfair review. A more graphic (and somewhat common usage) is as follows (apologies for vulgarity if it offends):
If you're going to fuck me, you could at least show me the courtesy of buying me dinner first.
(This is a play on words -- "to fuck" can mean both "to have sex with" and "to wrong someone/to mistreat someone.") This is something that a person who feels betrayed might say to their betrayer. E.G. disgruntled employee to boss, spouse A to spouse B upon being served with divorce papers without warning, etc... Again, that last one is pretty vulgar so I wouldn't use it in day-to-day speech, but it serves the purpose of showing how this phrase is used.
Best Answer
It's a slightly quirky idiomatic usage that even many native speakers might not be familiar with.
What the writer means is that these "incidents" form the basis of interesting tales that can be recounted repeatedly in after-dinner conversations by a skilled raconteur (by implication, such as himself).
Because people like to be entertained in this way after a (formal) dinner, the host / organisers of the event would be likely to invite the writer. And there's usually the implication that any such "after-dinner speaker" would be likely to actually be paid to turn up (in addition to getting a free luxury meal, whereas the other diners are probably paying for the privilege of being there).
Sometimes there's no suggestion of payment / free food. I dined out on that story for years might simply mean When I went out for a meal with friends, I often used to tell that amusing anecdote, even if the speaker always scrupulously paid at least his fair share of the bill. He might just mean that his ability to entertain fellow-diners ensured such evenings were a success, and/or that people invited him out to dinner for this very reason.
Note that this idiomatic usage often includes other prepositions...
All those (and probably more) seem fine to me, but others may feel different.