Why not just go through it in order? Make the new stuff a treat earned for getting through everything else. I’d start with the TOS pilot and go forward. She’ll either like trek or she won’t, jumbling it up isn’t going to make her like it more, but it may make her like it less.
Don’t impose your preferences for doing things in sequence or being a completionist on new viewers please. This seems really bad advice and likely to turn off more potential fans than pull them in.
OP’s question is about how to figure out how to engage someone in the franchise who seems to have her own specific preferences, and things that put her off.
I’m a viewer who first saw TOS as a small child when it was in first run, and everything in first run after that. It for others, there’s a whole range of shows for different tastes, best to figure out which one suits someone’s tastes and pull them into the franchise with that.
When we wanted to introduce our kids to the franchise, we started with TAS, then curated episodes from the other series. Like many tween, Voyager turned out to be ‘their show’ and it makes sense that Prodigy is strongly tied to Voyager. Our kids have moved onto other Trek shows and other franchises as they’ve moved through their teens. TOS, DS9 & Enterprise remain shows that they’ll watch occasionally. But one can never say that they’ve not liked Trek.