What about GR? Is the situation analogous? Can we find a combination of constraints that behave nicely?

Does generate a normal coordinate transformation? no

What about . no

So not a gauge transformation. Uncoordinated effort does not generate gauge transformation.

Many examples follow of first-class constraints that do not generate gauge transformations.

Bergmann and Anderson 1951 say sometimes one needs to take a Poisson bracket of a velocity term. Their motivation is unclear. For Pitts it is necessary to make all of this stuff go through.

Get a bad physical change with a secondary first-class constraint.

Can we find a team effort? Yes by combining the primary and secondary first-class constraints.

Now possibility of change re-emerges in GR. Not exactly clear from this, but at least door open.