First, the Star Trek Universe differs from the Real World.
The trekverse has an moral substrate, it has philosophical relevance. In the trekverse, Picard can afford to discard the only weapon that can postpone the complete destruction of his civilization, of the entire Alpha Quadrant - the thalaron emitter - because, inevitably, another option will present itself. This solution - The Caeliar - is ridiculously improbable.
In the real world, any warrior who would choose not to use such an wepon, in such circumstances, deserves to be called an fool and an traitor.
Why?
Because the universe in which we live is ruled only by blind probability. It sucks, but it is so. There will be no deux ex machina Caeliar to save the day. Only death.
I invite anyone who disagrees with me to read some history. How many wars were stopped by divine intervention? Because that's what the Caeliar, Q, Oraganians, Metron are - gods. I'm sure that, throughout history, many prayed to God to save them because they're the "good guys", because they believe in Him, etc. How many times were their prayers answered?
Picard made the strategically correct decision when he chose not to use the thalaron weapon?
I disagree.
The war was lost the second the borg sent 7400 ships in the Alpha Quadrant and the Allied fleet didn't have an thalaron weapon to destroy them when they exited the subspace tunnel - an disastrous mistake on the part of the allied powers. There was only one queston left: how many will survive?
Yes, in the trekwerse the good guys win so you have Caeliar or Q or whatever - but i'm speaking from an realistic perspective.
Picard miraculously gets another chance to use the weapon - let's say he uses it.
In the worst case scenario, the thalaron radiation would have destroyed only part of the borg armada. This would have bought the Alpha Quadrant powers a few months before they were annihilated.
A few months - a gift from God. Billions of refugees could have escaped in these months - humans, vulcans, andorians, klingons, romulans, etc. They could have fled through the bajoran wormhole, or ventured in the intergalactic void.
Best case scenario - all borg ships are destroyed. The subspace tunnels don't exist anymore and the Alpha Quadrant has decades, perhaps as long as an century, to prepare for the next invasion - as established in "Mere Mortals". Furthermore, the borg from the Alpha Quadrant can't contact the ones from the Delta Quadrant to tell them how to adapt to thalaron radiation or transphasic torpedos - not that it matters. If, after an century, the Alpha Quadrant still only has these weapons, it's doomed.
I heard on this board that the borg can't be defeated by conventional means, and unconventional ideas only work once.
As far as current Federation&all capabilities go, this is true. However, I can think of at least two weapons who should be effective against the borg - in other words, the collective shouldn't be able to adapt against them.
One - telekinetics. In "Gods of Night", the so-called "children of the storm" destroyed hundreds of thousands of borg ships

via telekinetic means - and the borg didn't adapt. And there are other established facts that suggest that such weapons are beyond the borg's ability to adapt.
Two - fight fire with fire. The borg adapt through trial and error - hardly an creative endeavor. You want an effective weapon against them - copy this remarcable military technology. They adapt to your weapons - you adapt to their adaptations. Species 8472 probably used something similar (evolution through trial and error is inherent to living organisms and 8472's technology was organic) - and they came very close to destroying the collective.
The Federation prides itself with its great minds, its creativity - and it would have an century at its disposal. Plenty of time to develop such weapons.
And the "aftermath" of the use of the thalaron weapon - an arms race between the Alpha Quadrant powers - an joke compared to the borg apocalypse.
It was repeatedly stated that the borg will adapt to thalaron radiation. This means, of course, that such defenses are possible - and that the Federation&all will eventually develop them, as they developed energy shiels to counter matter-antimatter explosions.
Not to mention that everyone in the Alpha Quadrant will be highly motivated to work together in order to prepare for the next, inevitable, borg invasion.
And nobody in their right mind will blame the Federation because it saved trillions, including their civilization.
Now - let's say Picard doesn't use the weapon and the Caeliar are either uninclined or incapable of stopping the borg. I know, in the idealized Star Trek universe, this will never happen - there, an moral, perfect solution will exist for every problem - not very realistic. Nevertheless, let's see what hapens:
The borg "sterilize" the Alpha Quadrant, killing trillions. And Picard has his hollow "moral" victory. Moral - Pfft - there's nothing moral in condemning trillions to death because you don't want to use an weapon, to kill the genocidal sociopats that are the borg.