Then how do LGA and DCA, neither of which have full AoE status, have large numbers of pre-cleared flights to Canada, Bermuda, etc?

(The answer as far as I know is that you're mistaken; there is no requirement that pre-cleared flights fly to port of entry airports. My recollection is that DCA can in fact handle non pre-cleared international arrivals on private flights but not on commercial flights. Sounds from what you're saying like PAE -- which started this small diversion -- is similar anyway. But I'm pretty sure that LGA has no international arrivals facilities at all. Yes, on any pre-cleared flight, USCBP can require further formalities or approval on arrival at the US port, but they very rarely do. At international airports with separate or limited international gates (eg SEA), precleared flights essentially always arrive at the domestic gates. I have no knowledge of AS's SNA experience, but the LGA and DCA experience is very different than what you describe. Maybe the LA port of entry or whoever handled SNA wasn't used to handling precleared flights like the New York and Baltimore ports of entry would be for LGA/DCA and messed things up?)