If light doesn't travel through time, how do we measure its velocity? Why does it have a finite velocity?

You don’t get it and it’s not your fault.
When it’s dark outside go outside and turn on your flashlight. How fast does that beam of light go from your flashlight?
Well, it goes at the speed of light.
The speed of light is constant, time is not. We are on a rotating disk that orbits the sun that orbits the galaxy that is moving toward Andromeda and nobody knows how fast we’re going.
Speed is relative in relativity. There is no one “point” that we measure our speed against! We just are where we are.
Einstein didn’t define time or light he defined something he called “spacetime”.
If you had a surfboard going at .9999999999% the speed of light relative to Earth, and turned your flashlight forward it would leave your flashlight at the speed of light for you. Here on Earth, if we had a powerful enough telescope to see you, you would be going very slowly and you turned on your flashlight. It would leave your flashlight at the speed of light for us.
When you got back to Earth you’d be young and we’d be older.