• Yes, the capacitors next to the toroid inductors are part of the CPU voltage regulator output filtering.
• You can get into the weeds of component specifications but generally speaking, if a capacitor is 1800 uF and 6.3V it will probably be better than what's in there now. Low ESR and high ripple current rating are also nice for the output filter capacitors of a switching regulator, since it's the job of these capacitors to deal with the triangle-ish shape of inductor current so it doesn't turn into a lot of switching frequency voltage ripple. Also make sure you're getting parts with the right lead spacing and physical dimensions! Having them sit flush on the board and fit next to each other and adjacent parts is always nice.
• Replace all those capacitors at the same time. If the obviously bad ones are in that condition, I wouldn't expect the ones that aren't split open to be any good either.
• The regulator can deal with a certain amount of output filter degradation. The performance (voltage ripple/noise, load transient voltage excursions) will get worse but as long as it's not so bad that the CPU glitches, it will boot. You just haven't reached the point of "doesn't work at all" yet.
• Measuring components in-circuit is tricky, there's a lot of other parasitic stuff going on that can mess with your readings. On top of that, a multimeter may be able to measure capacitance, but there are some aspects of bad capacitors that require an LCR meter (or similar equipment) to check. You should really just replace the not-split open ones anyways.

When you plan your recap, find datasheets of those original caps and map them on your board and make a list of them. You can't use any general use caps on VRM, originals are very low ESR capacitors with high ripple current capability. Or you can, but results are most likely not good and using incorrect capacitors may cause not only unstable system, but also hardware damage.

For some of these caps equivalent electrolytics may be difficult to find anymore, but you can replace those with polymers and use roughly up to half the capacitance of original capacitors (there are not much over 2000uf polymers available).

This is the infamously failure-prone KZJ series from Nippon Chemi-Con (along with KZG). I have attached the datasheet.
As CharlieFoxtrot mentioned, you will probably have to look for polymer capacitors because there are no modern low-ESR replacements that match their specifications.
In fact, the Panasonic FR series (a common choice) does not even come close.

In particular, your 1800uF/6.3 V caps with an 8 mm case size are rated at 12 mΩ ESR and 2350 mA of ripple current at 100 kHz.

in general, everything is visible in the picture.
Its cheap polymer Capxon PS capacitior, and its quite enough.
In those days there were no such loads to look for TDK Epcos industrial capacitors.

In the original it was like this

CharlieFoxtrot wrote on 2023-09-08, 18:12:

For some of these caps equivalent electrolytics may be difficult to find anymore, but you can replace those with polymers and use roughly up to half the capacitance of original capacitors (there are not much over 2000uf polymers available).

Is there any documentation of how that "roughly up to half" works? Does that mean I'm looking for 1800 / 2 = 900 uF, 6.3V polymer caps? Am I looking for tantalum or aluminium capacitors there?

watson wrote on 2023-09-08, 23:21:

This is the infamously failure-prone KZJ series from Nippon Chemi-Con (along with KZG). I have attached the datasheet. As Charli […]
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This is the infamously failure-prone KZJ series from Nippon Chemi-Con (along with KZG). I have attached the datasheet.
As CharlieFoxtrot mentioned, you will probably have to look for polymer capacitors because there are no modern low-ESR replacements that match their specifications.
In fact, the Panasonic FR series (a common choice) does not even come close.

In particular, your 1800uF/6.3 V caps with an 8 mm case size are rated at 12 mΩ ESR and 2350 mA of ripple current at 100 kHz.

Thanks for the data sheet and the values.

I found Panasonic's EEU-FR0J182 with 1800 uF, 6.3 VDC, 1.79 A ripple current and 28 mOhms ESR.
I also found Panasonic's EEU-FC0J182 with 1800 uF, 6.3 VDC, 1.44 A ripple current and 45 mOhms ESR.

I assume I need to match those as closely as possible with 12 mOhms ESR and 2.35 A ripple current?

Does "up to half" the capacitance mean I'm looking for 900 uF, 6.3 VDC, 2.35 A ripple current, 12 mOhms ESR? For ESR I assume lower is better, how about ripple current, is higher better there?

Any brand names / search terms I can use to find fitting polymer capacitors or pointers to guides/tables that I can use? (sorry if I'm missing something obvious)

thp wrote on 2023-09-11, 15:18:

Does "up to half" the capacitance mean I'm looking for 900 uF, 6.3 VDC, 2.35 A ripple current, 12 mOhms ESR? For ESR I assume lower is better, how about ripple current, is higher better there?

Ripple_Voltage=change of current * ESR.
I.e.
Capacitor have 10mOhm ESR, V_ripple = 50mV
Capacitor have 20mOhm ESR, V_ripple = 100mV = to high noise (ripple) = instability.
ESR should not be higher than the original capacitor.

Capacity refers to the stability of the regulator.
Too small and you'll get oscillations, heat and possibly magic smoke.
In practice, the capacity can be reduced by half (with the same ESR and ripple current) without instability of the entire VRM.

Therefore, the same cheap general use Lelon OCR 1200uF * 4V will be simply ideal.
10 mm diameter, 12 mOhm ESR and 5.5 Ampere ripple current.
http://www.lelon.com.tw/upload/prod/169381783974.pdf