The Chassidim HaRishonim were careful to calculate their marital relations to ensure that the birth wouldn't be on shabbos. Clearly this was not a requirement, just a minhag tov. The Maharatz Chiyus points out that this would only make sense within the framework of the Ba'al HaMeor but not the Ramban. There is a machlokes between the ba'al ha'meor and the ramban whether one can do a bris on shabbos knowing that after the bris the pikuach nefesh of the child will cause them to do a chilul shabbos such as heating water to help heal the wound. The Ba'al HaMeor says that one cannot do a bris knowing in advance that it will create a situation of pikuach nefesh and cause shabbos to be pushed off. Simply put, shabbos is dechuya for pikuach nefesh, not hutra. Whereas the Ramban holds that since at the time one will need to heat the water and violate shabbos, it will be a pikuach nefesh situation, there is no need to try to avoid it. The Ramban seems to hold that pikuach nefesh is hutra, not dechuya, so there is no reason to try and avoid it. The Maharatz Chiyus points out that within the framework of the Ramban it would make no sense to take measures to try and avoid a shabbos birth, it only makes sense to even have such a chumra within the framework of the ba'al ha'meor.
The Maharsha (chidushei agados) explains that the chassidim harishonim described in our gemara were really trying to avoid the double chilul shabbos. If the baby is born on shabbos then the bris will be a week later on shabbos, causing chilul shabbos twice in a row. They were trying to avoid chilul shabbos two weeks in a row. The Maharsha clearly assumes that the status of Mila on shabbos is tantamount to pikuach nefesh pushing off shabbos. It is something which must be done, but better to avoid. However, the Gr"a (Biurei HaGra end of Hilchos Mila) writes that avoda in the beis hamikdash on shabbos is hutra, pikuach nefesh is dechuya, and struggles with where milah should fall. He concludes that it is similar to avoda and is hutra on shabbos, not just dechuya.