It's worth noting that most cheap inverters/UPS are offline/standby or line-interactive types.
This uses a small relay to switch from Eishkom power to whatever battery power your inverter is powered by. Most good inverters will have a changeover time on that relay of less than 10ms.
(the best type of inverter to get is an 'online-UPS' which is permanently 'inverting' and avoids most problems of below)
Large home inverters, 3kw+ typically have a large capacitor or two to hold a current buffer during that failover time, depending on how much current you are drawing, the capacitor 'holdup' time will be different.
In the same breath, almost all of your home devices will have some level of capacitor inside of it as well, be it a computer power supply, phone charger, TV power brick, or even the transformer in a microwave can hold a small amount of current to allow for a small mains 'brownout' to not effect the devices operation.
Most PC hardware review sites, when reviewing power supplies will test this holdup time.
ATX Specification is set to be 17ms, which is enough to survive most good home inverters switchover.
Unfortunately, many PSU manufacturers tend to skimp in this regard and may have poorer holdup times, also affected by the age of the capacitors in your computer PSU.
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So if your holdup time is 11ms and your inverter takes 10ms to switch, all should be well, right?
Nope. Inverter changeover time is the amount of time it takes the relay to switchover, not the amount of time it takes to start drawing the correct amount of current from the batteries and converting it to AC power at the required amount.
As your line-interactive inverter switches from Eskom to battery, it needs to first energize the circuits that convert battery DC power to AC power for your computer power supply to use.
While energizing, it takes time for the inverter to build up to the correct power levels, and the the more load on the inverter, the more amperage is required, which results in a loss of voltage (Vdroop)
Once the inverter changes over, your PC PSU will also contribute to this with inrush current as it attempts to recharge the capacitors, which can make Vdroop worse from the UPS.
If you were to watch what happens on the AC output side of your inverter with an oscilloscope, you'd see something like the below, where Eskom drops, inverter kicks in, current is drawn from the inverter and it suffers a voltage drop. (I dont have an oscilloscope handy so my crude Mspaint image will do)
For those of us with sunsynk/deye etc inverters powering the house, it's actually this voltage drop that causes your LED lights to flicker during changeover, not the changeover itself. (Cheap LED lights typically do not have capacitors in them for holdup)
Devices in the household with a low power draw or good holdup time will survive the first loss of power (eskom) and continue to work through the 2nd loss of power (vdroop)
This Vdroop occurs everywhere in power delivery, from battery->inverter, then again at inverter->computer and again from computer PSU->CPU/GPU
If Vdroop at any of these levels exceeds what the tolerances are for any of the components, your device will shut off.
Once you factor all of these things in, most computer power supplies will handle a short interruption of power under full load, but may not survive the 2nd voltage drop of a cheap inverter under full load.
Here's a snippet of the RM1000X review, showing holdup time:
Holdup time there is 23.6ms under full load, under half load you could realistically expect ~38ms, under quarter load you could expect ~50ms, under 0% load you could expect *seconds* of holdup time, that's typically why your motherboard LED stays lit even when you've turned your PSU off.
If you want a longer holdup time, you need to oversize your power supply on your PC, if all your components draw 400W, get an 800W PSU, etc
Alternatively, investing in an 'online' UPS/inverter would shorten the time to operation, allowing a 'to-spec' PSU to achieve its holdup time.
Holdup time will also go down as capacitors age and lose their total energy capacity.
TL;DR: Capacitors in *good* computer PSUs can survive a changeover at full load (gaming), but not a changeover to a unstable power source (cheap inverter) - massively oversize your power supply so it's only at 50-60% use during gaming or get an 'online' type inverter paired with a decent computer PSU.
