All models with "LC" in their name were popular in schools. In fact, the LC line became education-only after some time. I believe the original LCIII (not LCIII+) was the last LC sold to consumers.
Schools liked the LC because it was inexpensive, could display color, and could run Apple II software with the Apple IIe card. Keep in mind many schools had massive libraries of Apple II software sitting around by the time the LC came out in late 1990. With the LC, they could buy new computers and not sacrifice compatibility. The slot for this card remained on subsequent models (and also showed up on the Color Classic, essentially a reworked LCII without the ability to run System 6).
The pizza box LCs kept selling well until the 520 came out, just in time for the 1993-1994 school year. The III was less popular than the I or II because:
A. The III came out in March 1993. Many schools don't buy computers in the middle of the year.
B. The 520 came out in time for 1993-1994, just in time to compete with the III for new computer sales. The AIO 520 was a better value on paper for many, especially if shipping was factored in.
C. The 520 was the same machine as the III.
D. Many schools had just purchased entire labs of IIs in 1992-1993 or Is in 1991-1992 and weren't ready to buy more.
E. (this one is very important): AIOs represented a better solution for computer labs due to the number of power outlets needed to run one. In schools that had actual labs (not labs retrofitted to old classrooms), two outlets per computer were common. Since CD-ROMs were taking off and an external CD-ROM would have added a third outlet, new power strips would have been needed--both an extra expense and one more thing to worry about during the course of a busy school day (believe me, I have seen rows of computers go out in labs because a student or teacher accidentally hits the switch). In retrofit classrooms, every power outlet is even more valuable.
F. The AIO is easier to move from classroom to classroom than the modular design--less wires, less parts.
By the time the 475 came out, Apple had to lower the price dramatically to get people to buy it over the 500 series. However, by the time the 1994-1995 school year came around, the AIO line was much more popular, especially with the Classics off the market by then. (Some schools preferred the Classics due to their smaller design, lower price, and increased portability). The 5000 series took over in 1995-1996 (but sold alongside the 500s for a while, with the older machines provided as value priced alternatives--remember the 580?)