I guess my question comes down to this: What is "better than Manuel with okay competence" based on here?

The traditional narrative may not be charitable to him, but it doesn't follow that means that it's wrong about him.

Though to be honest, the way this interests me as far as the what if: Say the 4th Crusade never disrupts his reign. What is Alexios actually attempting to do about the state's multitude of problems?

He's around fifty at the time as far as I can tell. Based on other emperors, he might have another ten or fifteen years at most (potentially longer, but not likely), and he's going to be followed by a son in law most likely - if they don't predecease him, and if no one overthrows him first.
He was an active campaigner and did actually seem to be doing a decent job of emperor-ing. If one looks at Kaldellis pointing out which stuff Choniates and many other contemporaries leave out of his known activities during his time as emperor. He spent 1195-1203 almost constantly working and working and had managed a stable-ish regime which just needed time to solidify properly.

His wife, and a number of others helped a lot in keeping the situation at home working. Not unlike the sort of partnership we see between Alexios I and his most trusted relatives. And it does seem like he was actually setting up a stable partnership with a number of leading families by way of strategic marriages which bound them closer to the center, and given what he'd inherited... Alexios III absolutely does deserve credit for what he was managing before the Fourth Crusade killed any chance of imperial stability being restored.
 
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