"Avian carriers can provide high delay, low throughput, and low altitude service."
"...only guarantees best effort delivery, loss of a carrier can be tolerated."
"...storms can cause data loss."
"Audit trails are automatically generated, and can often be found on logs and cable trays."
And probably most important:
"Security is not generally a problem in normal operation, but special measures must be taken (such as data encryption) when avian carriers are used in a tactical environment."
The only encryption method available is for the bearer to wear the ring, and Sauron has already cracked the key.
Avian carriers are only appropriate when it is feasible to have "persistent delivery retry, until the carrier drops." They're ruled out for this mission simply because there are no second chances.