Hi Zartonk, thanks very much for the response! Originally, that's what I assumed too (that teams didn't have crests on their shirts), but the 1973 team that played in the final definitely did. The photo you actually posted I think is actually from the 1975 final when Ararat Yerevan beat the Ukranian team Zorya Voroshilovgrad to win the Soviet Cup for the second time. I don't know why they had no crest. Perhaps because that was a back-up jersey that had to be worn so that they didn't clash with the other teams shirt color. In this video footage from the 1973 final you can see the jersey is dark and a crest is visible on the left side of the chest.
I think my best bet is to ask around, the next time I'm in Armenia. Perhaps I'll go visit the Hrazdan Stadium and see if I can get any info! I dunno...