1/36 is an estimate, it is not an estimator at all. An estimator is a formula based on data from the rolls.

It's an estimator in this case. A fixed number is an estimator too, it's just not going to be desirable in most cases. But single numbers are absolutely and unquestionably also valid estimators.

In any case, what I'm trying to get at there is that in estimator theory there is a concept of optimality for an estimator over a distribution.

Sure, a single number is a trivial example of a procedure/formula.

But an estimator estimates an unknown parameter from data (or in such a trivial estimator possibly without data) - and I believe this is central to the confusion.