Hidden fes and nonlinear operator linearization

I am a bit confused about the use of Hidden dofs in linear system solver.
In my understanding, hidden dofs does not contribute to the matrix and it is locally eliminated in the assembling step, and the associated component in the grid function can not be recovered.
This makes it more computationally efficient over static condensation where the condensed-out dofs are local dofs and can always be recovered after the global system solve.

But when I applied an HDG operator with hidden dofs to a nonlinear Navier-Stokes operator, but it to produce reasonable results. This is a bit strange as the operator works well on a linear opeartor.

I guess there must be something I am missing in the AssembleLinearization or Apply operator.
Any insights would be welcome.
I can provide a minimal example if someone is interested in investigating the issue.

Hi Guosheng,
which dofs are you hiding ? Only fluxes, or also the internal states ?
I think it's better to provide an example,
Joachim