Well, the 連用形 of most verbs can be used as nouns (called 'deverbal nouns') and/or noun elements in compound nouns.
「任せ」 is not quite like the examples above, in that it is not normally used as a stand-alone noun (though there is「お任せ」) , but it is quite productive as a noun element in compounds, such as 「運任せ」「力任せ」「他人任せ」 and even something like 「田中さん任せ」.
So 「学校任せ」 is just a compound noun phrase, nothing disturbing.
Now you may be able to see that 「学校任せにする」 is an instance of the pattern 「NP + にする」, meaning "(to) turn something into NP" or "(to) put something into a state of NP". (I guess the literal translation of 「子供の教育を学校任せにする」 would be something like "(to) put the education of children into a state of leave-it-to-schools.)
It should also be noted that there is a slight different in meaning between 「NP任せにする」 and 「NPに任せる」. The former often implies leaving most (or all) of a given responsibility to NP when you should take more (or at least some) part of it, whereas the latter carries no such implication. Thus:
「子供の教育は学校任せにすべきじゃない」 sounds closer to "We should not leave the education of children all to schools. (Parents should be responsible too.) "
「子供の教育は学校に任せるべきじゃない」 sounds closer to "We should not leave the education of children to schools (because schools are not to be trusted with such a responsibility)."