Not becoming one with the Reapers (another franchise anyway); am gaining Synthetic coded DNA so as to cease Galactic conflict. Temporary defeat for the sake of repeating the process is not a win; tis a stall tactic at best, and seemingly designed in a stall at worst...
There is no guarantee that there is a problem in the first place. All you have is an unverifiable claim by the Catalyst with no supporting evidence from the narrative. Does Synthesis cease all conflict, or just Synthetic vs Organic?
Also, even if it were just a stalling tactic, there is nothing wrong with that. The idea of the Galaxy rejecting the Reapers and coming together to figure out their own solution to the problem would be entirely consistent with the themes of the series.
Destroy solve the Reaper problem but doesn't necessitate the elimination of the over all synthetic vs organic problem.
In essence Destroy option treats the symptoms of the illness but doesn't actually address with illness. Control, Synthesis at least attempt to address both symptoms and illness in their own way.
That is not a problem. It's a claimed problem by the enemy.
How does Control address the "illness?"
Who says about "synthetic vs organic problem"? The A. I. that is about to be wipe from the galaxy by your weapon of super-mass destruction. The same A.I. that couldn't eliminate Leviathan, that couldn't prevent Protheans from altering Citadel signal, nor defeat of Sovereign or Collectors. Not the best advisor in my book.
I see what you did there, and I love it.
I'd never pick Destroy, but I'm skeptical of the Catalyst's argument. Shepard can contradict its arguments by making peace with the Geth and Quarians, and on a smaller scale by befriending EDI. On one hand, it has been around for a billion years and has presumably seen a lot during its observation of the cycles, on the other hand it's pretty clear (if you have the Leviathan DLC) that for all its age and knowledge, it's just a lazily-designed VI that can't break out of its original programming or see how its own actions are exacerbating the problem, and is stuck in a mental vicious cycle.
Narratively, the Catalyst can't just show up, claim to be an authority, and demand that we listen to it. Evidence has to be presented, if not by the character, then by the events prior. This didn't happen.
You're right that it is basically a VI, despite it's claims to the contrary.
Who said it could have prevented the signal alteration or the defeat of Sovereign or Collectors? You are jumping to a lot of conclusions of it's over all capabilities. It clearly has limitations of what it can and can't do.
Yes, it's ability to think is severely limited. It's far more bound by it's mission than EDI is.