Bicycle headlamps work fairly well,without that much extra effort by the rider.
And that's with incandescent lamps.
Hand operated "magneto torches" have been around for over 100 years.
They have a spring loaded "trigger" which when squeezed,rotates a permag generator.
Makes your hand sore after an hour or so,but if you don't need light you don't squeeze,which alleviates that.
My father had one years back which was made of metal & worked for years.
I bought a plastic one about 30 years ago,which also worked well till the plastic mechanism broke!!
A fairly large spring can store a lot of energy.
With the older rotary mowers you wound up a spring,which on the release turned a 1 or 2 cylinder engine against compression.
OK,it all comes out at once,but it doesn't have to.
Re :the falling weight.
Back in the day,the rotation of the lens of lightouses was driven by clockwork,using a weight which every few days was wound up to the top of the lighthouse.
As it fell slowly,it created enough power to turn the whole,quite heavy lens system.
All that is necessary is to put down a bore hole of similar depth to the lighthouse height,then use a similar mechanism to drive your "gravity powered light".