The phone is not handling the REGISTER properly at all, that was clear in the first debug that was claimed to be eaten up by HTML
This should be REGISTER -- 401 Challenge Reply -- REGISTER (most important, it's a NEW request) -- 200 OK (for valid register) or 403 Forbidden, etc.
Now based on that outline above, the first REGISTER request you send to the PBX has: CSeq: 101 REGISTER and the 401 Challenge shows CSeq: 101 REGISTER in it's reply. That is all good, that is correct. However, you will notice the second REGISTER that follows the 401 Challenge also has CSeq: 101 REGISTER <-- That is WRONG. That should be CSeq: 102 REGISTER.
A 401 Auth Challenge automatically forces the endpoint to reply with a new REGISTER message, this time with the auth/www digest inside of it (which holds the credentials)
The bottom line is the Cisco 79XX are garbage SIP phones. They are EOL, unsupported and have so many feature loses not being on the CUCM that in order to get some of them back, you have to patch the crap out of Asterisk and each new Asterisk updates breaks the previous patch.
So to recap, these phones are screwing up the simple REGISTER flow. This is not a FreePBX/Asterisk issue this is completely the 79XX.