You could think it's something like this:
[私が]それをしない
I won't exist as doing that
Grammatically, 私がない is the sentence at its core. それをし just modifies ない. It describes the particular non-existence 私 does.
それがしない
That won't exist as doing (it)
The last one is pretty odd. No wonder it is never used. It doesn't mean "This is not done" or "This will not be done".
Lesson 88 expands on the issue of を and が in たい sentences. I will briefly summarize it.
Using を instead of が gives the idea of a non-impulse desire. それを分かる. さくらを助けたい. パンが食べたい. 君が好き (which is romantic as opposed to the を version which is calculative or utilitarian).
それを食べたい
たい is an adjective meaning "desire-inducing" and it is modified by それをたべ. The subject it is not 私 but the thing/situation that induces the desire to それをたべ. For example, "the fact of not having eaten それ for a long time" may be a subject.