>is it safe to just short pin 14 (green power on wire) to ground at the connector and leave it like that?
No.
I've built over a dozen computers and servers, some in the $6-8,000 range. Also did tear-downs and rebuilds of several dozen workstations and a half-dozen notebooks.
I've never had to short anything to make it work. I've always followed the schematics from the motherboard + chassis + PSU suppliers. If it's a tear-down, I always make my own sketches and/or take digital photos at each stage.
Jumpers are used for diagnostic purposes. Wherever jumpers are required for assembly, you will find labeled/numbered posts and pre-fabricated jumper.
I don't know if any ON/OFF computers are made any more. I haven't seen one in quite a few years...a Pentium processor with 96MB of RAM, IIRC. All modern desktops, notebooks, servers, etc. come with a logic circuit. Push the hardware button (which you've tested already for continuity) and it engages logic for power up, power down, sleep, hibernate, etc.
This is controlled by the BIOS and OS. From your description, you may have messed up the settings in the BIOS, or left the switch in a confused state.
Try these:
1. pull the battery for the CMOS (BIOS settings). Short the jumpers that will clear all BIOS settings. See your motherboard schematic for the jumper numbers. They will be in close proximity to the battery.
2. Reinstall battery.
3. Boot (POST) and enter BIOS setup.
4. choose default settings ("conservative" or "fail-safe").
5. see if desktop will power cycle reliably
--reset soft-switch--works on desktops and notebooks with "confused" power buttons
1. unplug mains (AC) from wall
2. remove batter power (if notebook)
3. push down power button for 10 seconds
4. release
5. plug in mains
6. attempt to power cycle
7. if power cycles OK, re-install battery