At a school that large I could use two binders or the single sheet per child method. You would put the sheets in their files at the end of the day. At other larger schools, I've seen a different binder for each grade. One school with, I think it was 4000 students, a high school, printed the students individual 4 X 5 1/2 inch log on card stock and keep them in kind of shoebox shaped plastic bin on the desk. Each card has labels made up by the main office with the child's name and grade and, if lower grades, teacher; in the older grades, homeroom. When the card if filled there is another set of labels to make a new card. The cards are numbered also. Each grade had a different color - freshman green, sohomores blue, etc. At the end of the year, the students log went in their file.
To be able to provide a record for an individual day, I still endorse using a daily log. "Directory " information is allowed with multiple students names, so as long as you do not have the students put their complaint on the card, you have a record of who came in your office that day, what time, when they left and their dispositon, back to class or home.
Of course all this is moot if you can get the district to invest in health office software or if you can get a grant to fund it.