OK, so to sum up... Why are all DIY amps external? Well, they're not ALL external...
I've built a few subs. Some are pro-audio, and those use external amps, as is convention for pro-audio. I've built some car subs, and those use external amps for the same reason.
OTOH, I've built a couple powered speaker sets for desk / PC use. Those use plate amps, as is convention for powered speakers. (I even built my own amps, because hey... DIY, right?)
I haven't built any HT subs -- yet -- but if/when I do, I would plate-amp them if I were giving them to someone else, because most HT receivers only give you a line-level output, and the recipient is not likely to want to build a 19" rack just to suit this one sub. For myself, it would be decided based on what else I had going on. I like plate amps, and I like rack amps. They're both perfectly fine.
My current home sub is a commercial box I've had for some 10 years or so. Plate amp. Still works wonderfully.
Stepping away from the religious debate for a moment, there are pros and cons to each form factor, of course.
Plates are somewhat limited in their footprint, and are subject to mechanical stresses. I've taken to building an "amp chamber" in my self-powered sub boxes, so there's no real concern about air leaks. It's not too hard to flush-mount a plate with a router, but rounded corners can be tricky. I've never had any problem with ground loops, and I don't know exactly where you're going to find a "loop" with usually ONE connection to the AV system... (The power supply should be galvanically-isolated from the AC mains afterall.)
Racks open up possibilities for swapping the power-plant without modification to an enclosure, which is nice if you're prone to tweaking things. The market is more competitive, which means wider variety, and potentially lower costs. You (that is, the manufacturer) can actively cool a plate just as easily as a rack, if you're OK with fan noise. If you're not, the plate may win on based on having vertical fins exposed to air -- that, and I've never seen anyone put a glass door in front of a sub enclosure. So I'm not sure cooling is all roses and sunshine either way. Finally, depending on the amp, you may have to expand your connector vocabulary a bit. (XLR? Speakon? Balanced inputs? Whassat, now?) Not insurmountable, and Speakons in particular are actually quite nice.
Sure, there aren't too many plate amps that put out enough power to turn your driver's voice coil into a backup cook-top, and therefore practicality might be a deciding factor in a few cases. Otherwise... I'd say it's pretty much down to one thing: It's a little easier to build a box with one hole in it, than one with two holes in it*, and people are inherently lazy.
(*: Plus/minus one or two for ports, terminal cups, etc...)