You could configure BIOS/UEFI to power-on when power is restored (or return to last state). Or use wake-on-LAN, which quite a few consumer routers can send through their web interface.

Don't your servers already have BMCs supporting IPMI to provide full remote management? Often features like full KVM will require extra licensing, but remote power on is one of the most basic features and I've never encountered a BMC that didn't provide at least that much remote control.

Sounds like you might need a UPS/Battery backup for outages.

Some of those servers can have multiple power supplies for failover too. They also can have cards in them to power them on/off remotely as well as long as they have power.