Hi all!
I've had a 1200XL in the closet for the last 10 years, and decided it was finally time to get to work exploring the Atari world. Aside from borrowing a neighbor's 800XL for a few days as a kid, my experiences from the era involved mostly the C64, then Amiga.
Perusing photos, it seems the 1200XL came with (mostly, at least) 2 brands of capacitor--Elna (made in Japan), and NiCon. Mine had the latter, and from what little I could gather online, these are Chinese.
I tested the NiCon caps with an ESR meter, and found all well within in spec save the 22UF/16V parts. Of these, most tested near the threshold (1 to 2 ohms), but one was clearly defective at 45 ohms! I also noticed the "rotten fish" smell while desoldering some of them, the typical hallmark of leaked electrolyte. I replaced every axial cap on the board with Nichicon VX series (same values and sizes as the originals, made in USA), the large one Nichicon VR.
FWIW, the original Japanese parts on my C64's (Nichicon, Elna, Shoei, KYOEI) have all tested OK (ESR similar to new ones of the same values). My suspicion is that if your 1200XL has the Elna capacitors, they are probably still just fine.
After replacing the caps, I fitted an Ultimate 1MB, Side2 cartridge, and one of Simius' RGB boards (this thing is RIDICULOUS with my old Sony PVM, stunning picture quality!). Absolutely loving this setup so far. My 4 year old son and I have been playing "Rescue on Fractalus" together the last few days... what an amazing game!