Rabbi Sternbuch notes: “It seems from Rambam’s words [i.e., ‘how does he give thanksgiving and how does he bless’] that the halachic requirement of 10 in our case is actually two different requirements. First, that the blessing be recited before 10 men, and second – regarding thanksgiving – that the congregation of 10 offer thanksgiving and praise to G-d just like the person [saying HaGomel].”
Rabbi Sternbuch cites from the Rambam, explaining that the proper way to fulfill the requirement of saying Birkat HaGomel is with a minyan, excluding the person saying the berachah. The Chasam Sofer concludes similarly, writing that this is due to the rule of ‘B’rov am hadrat melech – In a multitude of [His] people is the King’s glory” (Proverbs 14:25). He also writes that we should not disregard minhagei Yisrael, age-old customs of our people. Thus, in his view, the minyan requirement is only l’chatchilah; in extenuating circumstances, there is room to be lenient.
Based on all the above, your rabbi, as the synagogue’s halachic authority, was certainly correct in making a rule that anyone who needs to recite Birkat HaGomel should do so with what he considers b’rov am – the large minyan on Shabbat.