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@LSpice, you're right; I was going to write that he either always does or never does depending on your interpretation. He rejects the notion of infinity, so your initial set of interest would be finite anyways from his point of view...
In some sense, Zeilberger either always proves statements by reduction to finite sets (depending on your interpretation) because he is an ultrafinitist and, in particular, he firmly rejects the very notion of infinity. For instance, see this opinion piece.
I assume that by Nakayama reciprocity, you mean the statement of Frobenius reciprocity for the induction and restriction functors? Then, is the Gelfand pair condition is equivalent to the Hecke algebra being commutative over any characteristic when G is a compact (or locally compact) group?
Ah, so it is actually in the equivalence between the Hecke algebra being commutative and the multiplicity one statement? Indeed, the use of semisimplicity (Maschke's theorem) is also present in Theorem 2 in Chapter IV of Lang's $SL_2(\mathbb{R})$ and in Proposition 6 here: mathematics.stanford.edu/wp-content/uploads/2013/08/….
