According to TIMATIC, the database used by airlines to decide whether a passenger's documents are sufficient, you can indeed use a birth certificate to prove your US citizenship when arriving by air, but as noted in a comment, you will probably have to identify yourself with a photo ID before being allowed to board the plane in Mexico. The TIMATIC text:
Nationals of the USA with a recommended proof of citizenship such as a:
- US birth certificate; or
- US certificate of citizenship; or
- US certificate of naturalization; or
- US Consular Report of Birth Abroad.
Nationals of USA with a NEXUS Card embarking in USA or Canada and entering Canada from the USA or US territories .
Nationals of USA with an emergency passport.
Passports and other documents accepted for entry must be valid for the period of intended stay.
Warning: if departing from the USA (regardless of any destination passport exemptions), a valid passport and/or accepted departure document is required by the US immigration authorities.
Holders of proof of citizenship issued by USA not holding a valid passport; will be granted entry upon arrival only after examination by a Canadian Border Services Officer if having satisfied the officer about their citizenship.
Given the uncertainty expressed in the last paragraph, I suspect that even if you had a photo ID such as a driver's license, the airline would not let you board.