The requirement $\delta S =0$ is a stationary action condition. It should be valid even for one field, say, for $a$. It has nothing to do with a gauge field definition, IMHO. Thus, one obtains the equations of motion for the field $a$: $E_a=0$.

In the variation principle the fields $a$, $b$, and $c$ are supposed to be independent and their variations too. Otherwise you have a system with constraints.

Originally, the gauge fields were called "compensating fields", find out why.

Briefly, some variables changes preserve the form (but not the solutions) of the equations: $a\to a',\;\; E_{a'}=0$. Invariance is only required for observable results of calculations, as a matter of fact.

(Worse, any reasonable variable change, even not preserving the formal form of equations, is good if it helps solve the equations.)