Out of their reach(es)?

Hi,

I'm a bit stuck on this particular grammar issue:

As a nutritionist, Jane was careful to keep the cookies out of her child's reach. OK.

As a nutritionist, Jane was careful to keep the cookies out of her children's _____.

Reach? Or Reaches?

Intuitively, any native speaker would choose 'reach', as reaches just sounds unnatural, but why would this be correct?

I will be coming over to pick up the patient's file.
I will be coming over to pick up the patients' files.

Are 'out of reach' / 'out of my/their way' / other similar phrases exempt from grammar for just being idiomatic or something?

Thanks!

Expert advice appreciated