The mishna on 65a explained that they would harvest the omer in large groups reciting responsive statements to show that they are not in line with the baytusim who had an agenda to make shavuos on sunday. The gemara 65a on the bottom indicates that the distortion of the baytusim didn't stem from their misinterpretation of the verse ממחרת השבת, rather they had an agenda to make shavuos on Sunday and therefore misinterpreted the verse.
Why did the baytusim want shavuos to be on Sunday?
It seems that they were primarily be creating a separation between the holiday of shavuos and the day of receiving the Torah. Being that the Torah never connects shavuos with matan torah, they had the ability to claim that the two have nothing to do with one another. If it were a celebration of matan Torah it must always be 50 days after the second day of pesach, but if the 50 day count only began from the next shabbos, it would not fall on the day of matan torah.
The Chasam Sofer (O.C. 141) raises the question - why didn't they want to celebrate matan torah? The Baytusim also believed in the written Torah so what was the point of creating a disassociation with shavuos? The Chasam Sofer explains that according to our tradition, the written Torah was accepted through na'aseh v'nishma, but the oral Torah had to be forced upon us כפה עליהם את ההר כגיגית (shabbos 88a). It doesn't seem right to celebrate a Torah that was forced upon us, therefore we celebrate the willful acceptance of the written Torah. But, the baytusim who didn't believe in an oral torah must have understood that he force was on the written Torah, so there is nothing to celebrate. That is why they felt that there shouldn't be a holiday associated with receiving the Torah. The approach of the C.S. strange because it is likely that the baytusim wouldn't have had the "tradition" that any torah was forced upon us and would have assumed that we accepted the "only" torah which was the written one by saying na'aseh v'nishma.
I would like to suggest another approach. The gemara 65a explains that in the conversation between R. Yochanan Ben Zakai and the Baytusim they made a claim that Moshe on his own tried to make shavuos back to back with shabbos so that klal yisroel can celebrate for two days. This indicates that their primary intention was to say that Shavuos was not a divine holiday, rather it was an institution of Moshe Rabbeinu. Why did they want to say this? I explained in my sefer Mayim Rabim (pg 205) that the events of matan Torah where Moshe added an extra day, was an implementation of the power of the Rabbonon to interpret the Torah and innovate law on a d'oraysa level. The Baytusim obviously couldn't accept that the Torah handed this level of power to the Rabbonon. They therefore would assume that Moshe's additional day was something he did on his own, without divine intervention. The Baytusim follow through with this conspiracy theory by saying that Moshe did things on his own, without G-d telling him to do so. They were adamant that shavuos should be on Sunday to indicate that it was Moshe's doing, not G-d, which would support the notion that there was never any power given to human beings at the time of matan torah.