In an earlier Mishna (86b) we learned that one is allowed to bathe the miila on Shabbos; a halacha which seems to be agreed upon by all. Is this the same case as the machlokes in our Mishna?
If ‘Mezalfin’ is sufficient for the newborn’s wellbeing, why does Rebbi Elazar ben Azarya permit bathing?
The Gemara says earlier on that Mishna on 86a that it is according to Rebbi Elazar Ben Azaria.
The second question is mainly according to Reb Yehuda Rabba Ber Avuha, where the Machlokes in th eMishna is about how much to allow. According to Rava, the Machlokes is simply about the extent of the Sakana of the third day.
Bathing is more efficient. Lehalacha, once you are Mechalel Shabbos for a Chole Sheyesh Bo Sakana we don’t evaluate each remedy to decide if this particular treatment would make a difference of life and death. This is what it means when it says עבדינן ליה יום חול. The Chachamim say that since Mezalfin is enough, just like we try as much as possible to minimize Chilul Shabbos while carrying and oher Melachos we can easily afford to minimize Rechitza. Rebbi Elazar holds that we should not refrain from the regular treatment.