In addition to access for the hierarchs and essential workers such as Artisan Flec (it is stated that the work Flec is conducting could have been achieved by members of the Decinarian math, but not in the time left before the ten-year apert, hence special dispensation being made for him and his workers), the Day Gate appears to have two roles, both directly related to maintaining the relationship between the concent and those beyond its walls:
Firstly, much as the Unarian math seems to provide a role in educating those extramuros, the concent seems to provide some kind of educational outreach to the general population, accessible via the Day Gate.
She headed straight for Grandsuur Ylma, whom she seemed to recognize, and began explaining that her father, who had died three years ago, had been a great supporter of the concent and liked to go in the Day Gate to attend lectures and read books.
Secondly, the concent (presumably via the Unarian math) seems to operate some kind of hospital for those extramuros. Conceivably depending on the status of civilisation outside the concent, this is could be one of the few places someone could receive evidence-based healing, and this must also help relations with those living beyond the walls.
... she had been kept in the Unarian math, which was more convenient to Physicians’ Commons.
The Day Gate was so lofty I didn’t notice we’d passed through it. An inlaid road of red stone, wide enough to drive two mobes abreast, veered to the right and plunged under a huge Mathic pile that I mistook for the Mynster. But this was merely their Physicians’ Commons, and the red road was a sign for illiterate patients and their visitors.