So, who was crazy enough to get adamant armor for every member of the party?

Funny you should mention that, because just recently I accomplished this on the PSX version (Final Fantasy Chronicles). I would LOVE to meet anybody who did this on the SNES version, considering there are no Alert items. Getting just one Pink Tail must be hell.

I only managed to get one on a file in which I only had two alerts. Dozens upon dozens of hours running back and forth in that room and at least a hundred Pink Puff fights before I saw "pink tail" on the screen.

Well, getting a Pink Tail with only 2 Alerts is quite impressive. I stocked up on about 300 Alerts when I was inside the Giant of Bab-il, and had only 30 left when I received my 5th and final Pink Tail. All my characters are pretty much impervious to any type of attack, thanks to the amazing Adamant Armor. However, I'm still trying to obtain other rarities like Glass Helms and Crystal Rings, with no such luck so far...

Where can I stock up on Alerts? I'm at the end of the game and just found out about this Adamant armor. Am I SOL since I'm not in the Giant of Bab-il part?

There's no convenient way to pick them up once you're at the end. If you have a lot of patience, you can keep fighting arachnes until you win one (very rare).