As I said earlier on:
I always tell the pax that atc have told us that our earliest take off time is....
Thereby nobody gets the blame and no lie has been told.
If I know what the slot is for, which I often don't, then I will explain that the reason we have a take of delay is due to poor weather at...
If you miss a slot can we blame you?
erm..yes you can, oh but hang on a sec, what if it was because the agents were late loading the bags, I might get a little precious and upset if you blame me for it!!!
Come on guys, it's not rocket science, a few little white lies or half truths never did any harm, I am sure that most ATC chaps will appreciate that saying "Sorry, Ladies and Gentlemen, we hae an air traffic delay of.." or something along those lines is not a personal attack but a plausible and as close to the exact facts as is required to tell the pax.
As I said earlier I do not advocate laying the blame at the feet of ATC if it is say a tech problem or something else like that.