"Out of battery" isn't really what we think it means. The phone shuts down (and refuses to start back up) when it still has quite a bit of battery left. If it truly allowed the battery to go flat, that would damage the battery, and you'd have a really hard time charging it again.

The amount of juice your entire phone needs to run for, say, a half hour can probably power just the bluetooth LE chipset for occasional wakeups for beacon-sending for... weeks? Months? (Completely made up time spans, but the BTLE chipset really does need a tiny amount of power for infrequent wakesups compared to the entire phone.) Certainly this can't last forever: if your phone is missing for long enough, the battery will eventually run down to a point where the battery controller won't let even the BTLE chipset draw any more power.

This definitely requires hardware, firmware, and software support in order to set up, which is why only Pixel 8 is supported so far. Clearly when they were designing the Pixel 8 hardware, they already knew this feature was in the pipeline, and wanted to be able to launch with a supported phone.