I think hybrid electric designs are 'coming' partly for that exact reason. Aircraft can takeoff and land using mostly battery power which would reduce noise considerably. Part of the motivation is also to increase the bypass ratio by running a multiple fans per set of engines. Like four fans and two engines. High bypass --> less noise.

I've wondered why catapult designs haven't been studied for airports, since they're used frequently for launching fighter jets from aircraft carriers.

It would cut down on fuel consumption, emissions and noise when taking off.

G forces and the fact that a lot of noise isn't from takeoff itself but just from the low flying that occurs near takeoff and landing?

They would put massive stress on airframes that are not designed for them. And (at a guess) they are probably far more accident prone than unassisted take-off, a trade-off that is not acceptable for passenger operations.

Airbus has studied that. We might see it eventually. But it would be tremendously expensive to build electric catapults long enough to keep G forces down to tolerable levels. And aircraft would have to be redesigned with a strong attachment point.

https://www.theregister.co.uk/2012/09/07/airbus_smarter_skie...

Do catapults on aircraft carriers cut down noise and fuel consumption? No because you still run the engines at max.

Ostensibly over a shorter period of time though.

Not at all, planes keep full afterburner until they reach a safe speed and level (varies by aircraft) and most planes are refueled again after the take off. The catapult just enables them to take off with serious costs for the pilot, plane and energy.

This approach is focusing on cruising, not take off and landing; the requirements are completely different, the solutions may be quite different.