You say you get no beeps without RAM, but when you do have RAM are you getting beeps? Those beeps are coded and can tell you what problem you have if present. I bought a Gateway desktop in '06 and had problems with the keyboard, monitor, and finally the mobo crapped out after just over 3 years, so I'm no fan of theirs. I've had much better luck with Dell, Toshiba, and HP.
Gateway All-in-one ZX6951 with Acer H67H2-AD motherboard, 4GB RAM powers on when I insert the power cord and then shuts down in about 3 seconds. If I pull the plug and reinsert nothing happens for about 2 seconds then the fan begins to whir and 2 seconds later it shuts off again.
If I reset CMOS by moving jumper it goes back to starting immediately upon insertion of power cord and again shuts down in about 3 seconds. After that it's back to nothing for a couple seconds when I insert the power cord and then fan whirs again and shuts off.
I have tried new power supply, removed RAM, HD, unplugged all devices, etc. all to no avail. When I remove RAM I get no beeps of any kind. I'm out of guesses.
When I reset CMOS and insert power cord and it starts immediately and remains on for 3 seconds versus subsequent insertion of power cord where nothing happens immediately and then fans whir and shuts down must mean something but I'm not sure what.