Hi Devin!
This is most likely caused by the capacitors on the motherboard. It's common that they fail and preventing the Xbox Original from booting. Your power supply should be fine.
They are under the power supply, on the right (in your photo). In your photo, I can see 3 of them which failed already next to the heatsink, close to the power connector. You can see the orange liquid coming out (dots) on the capacitors. And the other one is getting bad as well.
Replace all those capacitors... You need 5x capacitor 6.3v 3300uf.
You can buy those capacitors at local repair shops. Or you can buy them here: https://wiredopposite.com/product/original-xbox-capacitor-replacement-kit/?currency=USD
Good luck with the repair and let us know how it goes!