I just received a first class airmail parcel from New York and it took one additional day than the time it takes for a similar parcel to arrive from Los Angeles, so 22 days to get to Chile and 1 day to get to the coast.
I have received probably 9 USPS Airmail or USPS First Class Airmail parcels in the past three months or so and all have reached me and any additional delay than the 21 days to get to Chile can always be attributed to a holiday or a customs strike.
Each parcel had the mailing label directly on the box and from what I can tell, not one was opened by Chile Aduanas. Other stuff that gets put on these bare boxes are the USPS stickers and declaration, the Chile Aduanas stamp and the Chile Correos sticker. I believe the OP's mistake was to put wrapping paper on the box and thus all the sending and receiving information on that paper. This would be a very tempting target IMO for a lost mail scam / free gift for someone with few scruples.
Moral of story: DO NOT PUT ANY SORT OF WRAPPING PAPER ON USPS PARCELS TO CHILE, AFFIX ALL INFO FIRMLY ONTO THE BARE BOX AND JUST USE REGULAR CLEAR PACKING TAPE.
YMMV.


