Character Analysis
Like he does with the lucky, lucky pin makers, Adam Smith talks about schoolteachers so he can make a general point about all workers. In this case, he brings up schoolteachers to talk about how workers need financial incentive if they're going to be productive.
In other words, you can't just pay them the same amount no matter how well they do. And that's why Smith doesn't like the fact that teachers make the same salary whether they're good or bad. That's why he asks:
Have those publick endowments [and salaries] […] contributed to encourage the diligence, and to improve the abilities of the teachers? (5.1.61)
His answer is "Um, no," but he wants to explain why. In Smith's mind, people will get super lazy unless you do something to light a fire under them. As he writes:
In every profession, the exertion of the greater part of those who exercise it, is always in proportion to the necessity they are under of making that exertion. (5.1.62)
In other words, you can't let a worker get too secure in their income and job because they'll eventually get lazy. That's why he feels that:
The endowments of schools and colleges have necessarily diminished more or less the necessity of application in the teachers. (5.1.63)
But if he had his way, teachers would be paid based on how good a job they do at teaching the students.
For the reasons he's outlined, Smith prefers the university system to the public grade school system because he feels like universities do more to motivate their teachers to be better. As he writes,
In some universities the salary makes but a part, and frequently but a small part of the emoluments of the teacher, of which the greater part arises from the honoraries or fees of his pupils. (5.1.64)
In other words, the teachers rely for most of their pay on their students, which means they're motivated to make their students like them. So at the end of the day, Smith thinks that teachers should work for tips like waitresses. Just go repeat this argument to a schoolteacher and see if they agree. (Psst, they won't.)