AFAIK, the "protected" indicator measures that a small amount of current is flowing through the varistor under normal voltage. Just enough to show that is has not been open circuited as the result of fault current.
And yes, if the surge is small enough (in both voltage and current) the MOV will have enough thermal mass to absorb/clamp the surge without damage to itself.
If the surge is larger, the MOV will either fail by shorting (which would be apparent and an OCPD somewhere would open) or by open circuiting, in which case the protected indicator would go out.
But if you know that any surge arrestor has been subjected to a major surge, it would be conservative to replace it, since you do not have the tools to test it fully.