I became a Magister of House Telvanni, got the quest to kill Gothren, killed him, and reported back to Master Aryon, and he promoted me to Archmagister, but I didn't get any journal entry to that effect, nor was the quest marked as completed.

If I had to guess, it would be because I killed Gothren before the third phase of my stronghold was complete, but I'm honestly confused as to what trigger I missed.

It usually shouldn't matter when you kill Gothren, but depending on prior actions you may have run into a specific bug with advancement in House Telvanni:

[url=http://en.uesp.net/wiki/Morrowind:House_Telvanni#Telvanni_Master_Bug]http://en.uesp.net/wiki/Morrowind:House_Telvanni#Telvanni_Master_Bug[/url]

I haven't had any problems rising to the rank of Archmagister; it's just that Master Aryon's response when I reported Gothren's death did not update my journal or complete the Telvanni Archmagister quest. I still got advanced from Magister to Archmagister.

I suppose I could just type "Journal HT_Archmagister 100" into the console, but that seems too much like cheating to me.

For Bethesda games using the console like that is more akin to "fixing a rare issue" as long as you've satisfied the quest requirements.

All I had to do was click the "Magister" topic with Aryon twice. The first time, he promoted me to Archmagister but didn't update my journal, but the second time he updated my journal.

Even though this one was resolved in-game, I have to agree here - Using the console to fix a bug isn't cheating, since bugs are "outside of gameplay"...Even for Bethesda games.

For that matter, if you're playing Ironman, I don't consider it cheating to reload if you die due to a bug...Although for fairness, you should also reload if an NPC you're fighting dies due to a bug.