Let's look at the mandate from the Leviathan:
Leviathan: ... we created an intelligence with the mandate to preserve life at any cost. As the intelligence evolved, it studied the development of civilizations. Its understanding grew until it found a solution. In that instant, it betrayed us.
The intelligence has one purpose: preservation of life. That purpose has not been fulfilled. It directed the Reapers to create the mass relays - to speed the time between cycles for greatest efficiency. The galaxy itself became an experiment. Evolution its tool.
Shep: Will it ever end?
Leviathan: Unknown. Until the intelligence finds what it is looking for, the harvest will continue.
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And compare that with how the Catalyst explains its job description:
Catalyst: I control the Reapers. They are my solution.
Shep: Solution? To what?
Catalyst: Chaos. The created will always rebel against their creators. But we found a way to stop that from happening. A way to restore order.
...A construct. An intelligence designed eons ago to solve a problem. I was created to bring balance; to be the catalyst for peace between organics and synthetics.
When they asked that I solve the problem of conflict, they failed to understand they were part of the problem themselves. The flaws of their organic reasoning could not perceive this. They lacked the foresight to understand their destruction was part of the very solution they required.
... they created me to oversee the relations between synthetic and organic life. To establish a connection.
Reapers harvest all life: organic and synthetic, preserving them before they are forever lost to this conflict.
... new life, both organic and synthetic, can once again flourish.
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Whether there was more to the Leviathan's mandate than "preserve life at all costs" or the Catalyst refined and re-interpreted it is unknown.
Hmm, I don't think so.
Destroy doesn't solve the problem, yes, but it removes the immediate synthetic threat and gives organics a chance. "Clearly, organics are more resourceful than we realized" - the Catalyst acknowledges that the organic chaos is capable of feats it didn't predict. They managed to build and dock the Crucible. And if they fail to make peace, Crucible is always an option, just use it again and start anew 
Control is kinda the same, except with the added benefits of Reapers still being present, helping to recover and able to intervene if necessary.
Synthesis is, according to the Catalyst, an ideal solution. There are no un-synthesized species, the wave even affects trees 
A couple of things:
1) You're distilling the Leviathan's mandate and the Catalyst's description of its role down to simply this: Make sure synthetics don't wipe out organics. That might actually be what the Leviathan needed / wanted, but it does not fulfill the mandate or the Catalyst's stated interpretation of its job. It does not ensure the preservation of synthetic life, and tosses away all those previous harvests like so many expendable Cerberus lab rats. It only ends conflict by annihilating one side of that conflict.
2) The synthesis wave affects all species currently in existence. Others may evolve or be created that would not be synthesized.
Since destroy deliberately targets and kills synthetics it seems to be tailored to the brat's worldview. If it was solving Shep's problem it would just destroy the reapers.
I thought that was a technological shortcoming - because the Crucible had taken some damage or it just wasn't able to target Reapers without including other synthetics.
The Catalyst seemed to me to be equally sympathetic to both forms of life.