My SE30 came this way but I didn't realize it until I powered-up after re-capping and was greeted by arcing inside the tube and some very alarming buzzing sounds!
I replaced the CRT and the dead driver transistor on the neck board, and all's happy now.
The ACBel PSU on my G4 failed several years ago and I tracked down and replaced the failing cap thanks to threads over at badcaps.net. Search or start here for guidance...
Homebrew is essential for all sorts of opensource dodads missing from modern MacOS. in addition to hfsutils, I use telnet, wget, qemu, minipro, minicom, jq, ... and I could go on and on.
And there's Tigerbrew for PPC Macs but it's a bit of a tease to install.
In this case, the reverse-polarized tantalum draws a large current and pulls the -5V rail down (typically to around -2V). So there's not enough power for a pop, but you'll find that printer and modem ports may not work since they rely on the -5V supply.
The debate about C22 crops up from time to time. Here's a thread I started earlier this year after seing a failure: https://tinkerdifferent.com/threads/lciii-c22-polarity.2708/
The silk-screen and the original manufacture is wrong. No fault seems to occur using aluminum caps. But tantalums will...
Be aware that you've installed the infamous C22 reversed. Since you've used an aluminum cap (as per original) you shouldn't see a problem -- but if you'd used a tantalum, you'd would have. A reversed-polarized tantalum here will sink a high current and cause the -5V line to droop (or worse).
An alternative I've used dozens of times is a dental probe. A fine and rigid stainless steel tip chases solder out of through holes easily. Heat should be applied to both hole and probe.
I'd check the power switch contacts and deox if required.
And does the cold-start power-on happen with a Mac PSU? It may be a quirk of your ATX PSU plus adapter.
It perhaps goes without saying, but the IIci startup circuit is identical (modulo labeling) .. so you can compare the IIcx with that. And you can of course swap PSUs between them.
I think the issue is that they're not explicitly labeled but they're the reverse of their neighbors that are. So it's confusing.
If you're installing tantalums, at least putting these in backwards won't cause fireworks! Unlike the LCIII's infamously mis-labeled C22.
They are unused gates. It's similar for the IIci too. The startup logic was misfiring on one of my IIci's and I considered substituting an unused gate (turned out to be a near short elsewhere).
Note the caps in this circuit need to be good for the power-on/-off to work; they're used as RC...