Actually, belts would be no better than lead screws in this regard, and perhaps worse due to backlash between the belt teeth and the pulleys.
A step change is a step change.
With FT's 8mm pitch lead screws (8mm of vertical travel per revolution) and 1/16 micro-stepping using the A4988 drivers, the basic motion is 0.0025 mm per micro-step.
However, micro-stepping is not an exact position driver despite the common wisdom.
It relies upon balancing magnetic fields to achieve some compromise position based upon the residual torque and drag.
So for full steps between the motor's magnetic pole positions (where the rotor and stator poles are aligned) on a standard 50 pole stepper (200 full steps per resolution), the resolution is 0.04 mm per full step.
If you do a 2:1 pulley reduction as you have done, it becomes 0.02 mm/step.
And if you then go to 0.9 degree motors (100 poles, or 400 full steps per revolution) it becomes 0.01 mm/step.
If you are running 0.25mm layer heights and the standard setup, there would be a 0.01mm "error" every 6 full steps.
However this is non-cumulative, so....... ?
Finally, if you went to a 2mm pitch lead screw (2mm per complete revolution) and your motor and pulley reduction you would reduce that to (wait for it.....) 0.0025 mm/step (full steps that is).
Which is coincidentally 100 full steps for a 0.25mm layer.
Now you are talking accuracy.
Oddly, I am not sure why the 8mm pitch lead screws were used to start with on the Z axis.
2mm (single start) makes so much more sense in this application.
The Z motion is just not that fast, nor does it particularly need to be all that fast, I'd think.