The Rambam in his pirush hamishna explains that dayo is an accepted concept that everyone must agree with since it is learned from a pasuk and is d'oraysa. However, R' Tarfon and the Rabbonon argue about the application, whether we use "dayo" to destroy a kal v'chomer or just to limit a kal v'chomer. That is why the gemara cannot tolerate the possibility of there being a tana who rejects "dayo" even when it is just limiting the kal v'chomer, since it is a din d'oraysa.
Why does R' Tarfron hold that if the "dayo" would destroy the kal v'chomer we would forgo on the "dayo" in order to maintain the kal v'chomer?
Rashash explains that without the pasuk by miriam we would not limit the kal v'chomer at all because the entire logic of the kal v'chomer is predicated on the realization that the 'chamur' has chumros over the 'kal'. But, it is a gezeiras hakasuv that we say "dayo". R' Tarfon holds that since it is a gezeiras hakasuv, 'eil licha ela chidusho' - just as by miriam the dayo is just limiting the kal v'chomer, we can always only use dayo to limit the kal v'chomer. The Rambam also seems to understand along these lines. When we were given kal v'chomer as one of the 13 middos, it was meant to be used whenever possible to be mechadesh dinim. I would add that it is as if we have a mesorah telling us to make every single kal v'chomer that is possible to make (unless there is a pircha). Therefore, if we would destroy a kal v'chomer based on dayo, the Rambam says: יהיה זה הקל וחומר לשוא ולא יוסיף לנו ידיעת שום דבר - that is why we forgo the dayo and darshen the kal v'chomer.
Perhaps this explains Rashi in Succah 30a who holds that every one of the 13 middos requires a mesorah, but kal v'chomer doesn't. Why? The answer is that kal v'chomer is not really an exception, rather everything always needs a mesorah. But the mesorah by kal v'chomer is that every single kal v'chomer that can even be made without a pircha, is true. Even when dayo is applicable it will still not undermine the essence of the kal v'chomer.
On another point, the Pnei Yehoshua asks how can the dayo by miriam be the source for dayo of chatzi nezek. Perhaps the concept of dayo only applies when we try to create a din that we don't find a precedent for in any other case, such as being locked up for 14 days. But, since we find a din of nezek shaleim in reshus hanizuk by shein and regel, maybe dayo will not limit keren in reshus harabim to chatzi nezek? He points out that according to Tosafos the 14 days are also not arbitrary, rather they are sourced in the relationship with Hashem being double the father and mother, yet we say 'dayo' to limit the kal v'chomer, so here too we should say dayo.