I assume Core Watchdog may be different from System Watchdog and may only trigger when the CPU itself is unresponsive
Yes, that's what Core Watchdog is.
are there even system watchdogs on gaming motherboards?
Win has several built-in watchdogs in place. Same with GNU/Linux distros.
Best example: When program hangs/freezes under Win, watchdog timer is started and after certain time has passed, pop-up appears, telling that program is Not Responding or taking too long to respond. And gives you two options: either terminate the program (end process) or keep waiting (in hopes of it managing to clear the freeze on it's own).
There are watchdogs even on kernel level. E.g getting a BSoD usually shows the error info and after displaying it, OS starts to count down (~15 seconds) before automatically rebooting the entire system. Or asks user to reboot the system manually, without automatic reboot timer.
Now, i don't know software watchdog intricacies that well, but as far as i know, each and every program needs to have their own watchdog and timer, specifically coded for the program in question. Since some programs, for their normal operation, may need more time to respond than others.
Most of the times, when OS freezes, the OS itself isn't actually the one who froze, but instead a specific program froze. And when that program has highest priority, it overwrites other program inputs and makes using the rest of the OS impossible. Thus creating the situation of OS itself seemingly froze.
Though, there are kernel level commands to overwrite program priority, to remedy the situation. Most well known one is Ctrl + Alt + Del. This security feature brings up means to kill the frozen program (via Task Manager), to resume normal operation and use of OS.
As far as watchdog chip on MoBo goes, those aren't a thing on consumer hardware. Those can be found on server MoBos though. That usually only where server has only 1 task to execute and when software freezes, the watchdog chip on MoBo can execute the hardware restart.
On consumer MoBos, this watchdog chip on MoBo will do FAR more harm than good. Since people use their PCs for varying tasks, and not just single task, it is impossible to define the timeout for the chip accurately.
For example; let's say there is a watchdog chip on your MoBo and it is configured to time out and kick in once it detects that any program has sat still for 30 seconds. Now, if that program that has sat still for 30 seconds actually froze, then watchdog kicking in would reboot the system and clear the freezing.
But if the program that seemingly has stalled is actually computing (e.g bulk reorder of files), the watchdog catches it sitting still 30 seconds and will kick in, rebooting the entire system, despite system not being in the frozen state at all.
I want my PC to automatically reboot when Windows hangs.
You need to figure out if the OS actually froze or did one program froze.