Why does everyone keep using torsion balances for measuring the gravitational constant? Most of the experiments aimed at measuring the gravitational constant use very complex setups involving suspended balls (Cavendish-like experiments).
This is however not exactly the ideal setup – and in fact Big G is one of the least known constants. The biggest problems with the Cavendish setup are:

*

*The forces are really tiny

*Every object in the room influences the experiment (including the scientists walking in the room, whose mass is often larger than the test masses used)

*A sphere is not the ideal shape for measuring a force that decreases with the square of the distance; a plate is.

Why does no one use the following setup?
A $1\,\mathrm{m} \times 1\,\mathrm{m} \times 0.1\,\mathrm{m}$ lead plate is at rest on a balance.

After its weight is accurately measured, another similar plate is slided right under it.

If the distance between the two plates is within $1\,\mathrm{cm}$, the first plate should become 0.35 grams heavier, which is well within the range of high precision balances.
Or, even better, using liquid mercury instead of a sliding plate should solve all the practical issues related to moving heavy weights.
EDIT
Apparently someone has attempted to perform a similar measurement using 7 tonnes of mercury (see this article for more details).
 A: The Cavendish type of experiment used the time period $T$ of the oscillation to determine $G$.  Disturbances from outside could be made insignificant.

(from here)
$$G=\frac{2\pi^2 Lr^2\theta}{MT^2}$$
where $\theta$ is the angle that the bar turns.
The experiment achieved an accuracy of $0.6$ percent.
The method in your question does give a change in force $\Delta F$ of about $8.5\times 10^{-3}$ Newtons assuming two weights $10$ cm apart of the same mass of the lead plates.  As the mass is spread out let's estimate  $\Delta F = 5\times 10^{-3}$ N.
$G$ would be determined from $$G=\frac{FR^2}{M^2}$$ where we could assume that the distance between the plates $R$ and the mass of each plate $M$ were known to within
$0.6$ percent.
However to improve on the Cavendish experiment, you would have to measure the change in weight of the top lead plate to $0.6$ percent of $\Delta F = 5\times 10^{-3}$, that's $3\times 10^{-5}$ Newtons.
The lead plate weights about $11,000$ Newtons, so your looking for a change of about $1$ part in $370$ million.
As mentioned in the comments, there are balances that can weigh very light weights, but not a heavy weight and also detect such a small change.
It might be possible to set up a system of rotating masses e.g. 4, that pass under the top plate and try and generate a resonance and look for that - but then you'd be back to a Cavendish type experiment on a bigger scale.
If the scale of the Cavendish experiment improved the accuracy, there would have been such experiments by now, so it seems that increasing the scale doesn't improve the accuracy.
According to this website in 2018 there was an attempt to improve the accuracy using 'atom interferometry' but groups were also still using the torsion balance method and it's still regarded as the best method.
