Would the moon be a better launch site?

Yes, in fact people are estimating that we will start seriously attempting to get a base on the moon around 2060. I don't know how accurate those estimations are though.

In some ways, it would be a much better site for launching. No atmosphere means payloads would not require aerodynamic fairings, and there would be no losses to drag. Additionally, the lower surface gravity means the rocket would need less delta-v (change in velocity) to escape to orbit.

In other ways, it is a lousy launch site, at least currently. There is currently no way to manufacture propellant on the Moon. At the moment, anything you need on the Moon must be brought with you from Earth, and that's even more expensive than just launching from Earth.

Well, yes, but you have to get everything to the Moon first. That means effectively fighting your way out of two gravity wells instead of just one. The better plan would be assembly of a spacecraft in Earth orbit and sending it off from there.

Well, ..., yes it would. However, it doesn't make sense to send rockets to the moon just to send them agin into space. You need the ship building infrastructure there first. Which means we need to be there, build the factories to produce materials, and do the finishing there. If we started today wit would take the next 50 - 100 years to develops such infrastructure.

you have to get there first

but yes, having a base on the Moon has been studied

facepalm