Hi, I'm a gainfully employed translator of Japanese and linguist.

Short answer: the sentence is fine as it stands, although your alternative is equally grammatical (if less idiomatic - "学校へ行かない" gets about 1/4 of the ghits of "学校に行かない" by my count, and personally I'd never use へ).

Long answer: Japanese draws a distinction between verbs of A→B movement (e.g. 行く to go; 引っ越す to move house; 移動する to move/change location) and verbs that describe the manner of motion (e.g. 歩く to walk; 走る to run). The A→B verbs can happily take an indirect object complement (a に phrase) as the destination, because there's really no other possible meaning. You can't "go for school", as you put it.

Verbs of motion, on the other hand, are less flexible. These can't take an indirect object complement, and need a more expressly destinatory case (think へ、へと、まで). The reason for this is that a change of location is not implied in the verb, however counterintuitive that might seem to an English speaker.

Interestingly, motion verbs can take a direct object (を), such as in 街を歩く 'to walk around town'. Another indication that they're not strictly destinatory. Also they can take adverbs that further qualify motion, such as ぽつぽつ歩く 'to dawdle/mosey/toddle'(? tough to translate), whereas 行く can't. A→B verbs can however take adverbs that qualify the speed, like ゆっくり行く 'to go unhurriedly' - but this is different from ゆっくり歩く 'to walk in a slow manner'.

This is my first ever comment on HN so I've tried to be as informative as possible... If you have any counterexamples I'd be happy to try to explain them.