I think pretty much everything turns off with no input voltage.
UPS are good for:
- Automated graceful shutdown/hibernate during a power outage (or just keep running if power is out for only a few mins)
- Boosting power during a brown-out
- Disconnecting your equipment during a surge (switches to battery)
- Line conditioning
- Sacrificing itself to save your equipment when lightning hits your tree (true story--six of my neighbours lost equipment from that; I did not; they all bought APC UPSes after that)
- Peace of mind and not having to worry about unplugging equipment during heavy storms/bad power instances
Without knowing how your stabilisers are built, I'd wonder if they filter sags/surges that would damage a complex PSU (or other sensitive electronics) vs. keeping your household appliances from frying.
Realistically, line input would be the last thing I'd suspect if a motherboard had power problems; I'd look at the PSU itself first, then its load (PC components, fans, etc.).