I also have a bunch of computers
of various vintage -- Amigas being my favorite. Other than leaky, corrosive barrel-style batteries on every single machine that had those (get them out of there now!), I have had leaking caps on three machines. Two were AGA-era Amiga and one is a Mac SE/30.
My original Amiga 1200 (purchased new in February 1993) lost its sound because of leaking caps. At the time AmigaKit was selling replacement motherboards so that was the easiest fix. A1200 with second motherboard is still running stong.
My Amiga 4000 desktop (purchased used a few years ago) had bad battery damage but it was leaking caps that eventually disabled the motherboard. I bought a replacement used motherboard from an A.org user and it too had similar damage from leaking caps. It looked a little better, so I sent it to the man in France for some solder surgery. It's working just fine now, but not without some serious money being thrown at getting it going again.
The SE/30 stopped booting but I cleaned up the cap leakage as best I can. Still boots, but sound is very diminished. I have a kit of replacement caps, but they are too tiny for my terrible soldering skills.
My ECS and OCS-era Amigas have never had leaking cap problems -- just batteries. But with AGA Amigas, I'd keep an eye on those motherboards.
For Blake's purposes, I think an Amiga 1200 in the wedge with a new 030 accelerator and Indivision for a nice WHDLoad machine is a good idea. Or reduce the A2000 expansions to the point where it is a reliable, nice WHDLoad machine. AmigaKit's PCMCI network card offerings will get you online for IRC.... or just stick to the Sam for that. (I have a Sam440 and an X1000 and love both dearly.)
My own current classic Amiga project is a 2000/060 that has a MegaChip, Indivision ECS, Deneb (machine boots from USB sticks) and Video Toaster and Flyer cards. It has turned into a FrankenAmiga for sure, but I am having fun playing with the Flyer for edititing standard-def video footage.
I am on IRC with my A1200 right now, and I like to keep that machine simple. I do not consider it a FrankenAmiga. But I like both approaches.