2007 Prius won't light up

11.something under load is normal. There is a pretty substantial current draw, and any battery's voltage will decrease more the greater the current outflow. In our car if the start button is pressed once without a foot on the brake the voltmeter will show a number somewhere in the high eleven range, like 11.7 volts. And that isn't even really starting up the car fully. <10 under load is voltage collapse, in this or especially an ICE vehicle with a starter. Between 10 and 11 is a bit of a grey zone, the car might or might not start.

 

Could you take a video of what's going on in the car during startup?

It is necessary to understand at what stage you have a problem.
Starting a hybrid car is quite a complicated thing, it does not happen like in a regular car (insert the key, turn the key and the starter starts to spin the engine).