I agree with PC Hoarder Patrol, I also don't think the no-POST issue is from not having a 6p AUX cable connected... but you can actually check if you have a multimeter. 😀
IIRC, those 6p AUX cables are comprised of 3 extra grounding wires, 2 extra 3.3V (orange) wires, and one extra 5V (red) wire. With that said, if you set your MM to continuity and you can see that the 3.3V lines on the 6p AUX connector have continuity to the 3.3V lines on the ATX connector, and same with the 5V line... then the 6p AUX connector on your board is really only extra redundancy to provide more current... probably for things like AGP Pro, as mentioned above... which is not present on your board. On the other hand, if they are not connected, then that would be an issue as probably the RAM or chipset regs may not be getting power.
So start with checking the line connections on that 6p AUX connector. If all is good there, then post the CPU sSpec #, RAM type, and PSU brand and model, as suggested above.
Also check your CFR2032 CMOS battery's voltage. Some OEM boards do not like to start properly when those are dead / exhausted. In the case of Dell mobos from the latter P4 era, the auto-start is normal when CMOS settings have been lost (due to bad or missing CMOS battery), but the boards usually POST fine.
While at it, also check all of the board jumpers and even remove and reinsert them a few times, just to make sure they make good contact. On that note, do the same with the RAM and CPU. Sometimes contact oxidation may form when the hardware has been sitting around in storage for very long.
I don't think the issue would be cap-related - the motherboard's CPU VRM appears to use UCC LXZ caps on the input and Sanyo OS-CONs on the output, so those should still be fine. The smaller caps around the RAM and chipset, especially those near hot(er) regulators may be at slightly more risk of being aged from heat... but I still find it unlikely. I've removed a bunch from similar scrap Dell motherboards, and I don't recall running into any that were failed. Maybe a few were just starting to get marginal, but that usually wouldn't be enough to cause issues.