This is a great question and alas, the answer is remarkably like what we're seeking today: each school/district/city handled it a little bit differently.

One common reason schools re-opened in 1919 was that the teacher got better. I found multiple instances of a short notice in a local paper announcing that school would reopen or had reopened following the Christmas holiday because the teacher no longer had the flu. One listing read simply:

Centennial school in Hamilton township opened on Monday. The teacher, Miss Minnie Coble, has recovered from the flu.

Most of the re-opening notices that I found appeared to be clustered around January, 1919 but there were multiple instances where a school announced re-opening in the Fall of 1918 after a short closing and a second re-opening announcement appeared in December 1918 or early 1919.

many of the pupils live out of town and did not hear of the reopening of the school in time to come this morning. He had several telephone messages of inquiry as to the reopening and tomorrow will see an increase over the first day's attendance and, it will not be long, he predicts before the average attendance which today was more than 8O per cent will be back to normal.

Some schools did try to stay open during the pandemic but faced resistance from teachers. In November 1918 while the Anne Arundel County (Annapolis, Maryland) County Health Advisor was announcing that schools were going to open, a group of teachers in Pennsylvania refused to go to work as some of their colleagues had gone to a conference and contracted the flu. By August and September 1919, Spanish Flu had pretty much burned itself out and schools opened normally.

That said, there was another major part related to the re-opening of schools. From a previous response on the Spanish Flu and schools:

One final note about the impact of outbreaks on school children. To us in the modern era, it can be overwhelming to look at the statistics around the Spanish Flu (195,000 American deaths in just October 1918) and have difficulty connecting to the human impact. While looking for articles about schools' responses to the Spanish Flu, I came across dozens of funeral notices in local papers for children who died, even some from families that lost multiple children in one week or day. School can sometimes feel like an impersonal space but those notices serve as a reminder that the adults working in them have long tried to do what they feel is best for the children they're responsible for, making the best decisions they can with the information they have.

School, especially in smaller communities, were faced with dealing with the impact of students' who had passed. In some cases, school leaders advocated for whole-school assemblies that served as a form of public mourning and flying the school flag at half-mast. Other school leaders, and teachers, advocated for a chin up, no acknowledgment approach that meant no discussion of students - or teachers - who had passed. So while schools then were different, they were very similar to today's schools in that the adults around the children were doing their best to keep the children in their charge safe, healthy, and learning.