This app was mentioned in 16 comments, with an average of 1.69 upvotes
After doing this for a long time I have come to the conclusion that no "single" way is the best way. Take a little from everyone and fit it in to what works best for you and how you think.
If you get a chance read Terry McCarthy's book on cross. He's a lifelong PD and while he incorporates much of the Pozner and Dodd methods (and the 3 rules) he does it from the perspective of a defense attorney.
As for actually prepping crosses, for the last 5 years or so I have been using the Simple Mind app. https://play.google.com/store/apps/details?id=com.modelmakertools.simplemindfree&hl=en
Rather then thinking about cross in a linear fashion, the simple mind app allows you to create flow charts so you can pivot to any topic without shuffling through papers and perhaps losing your place.
https://www.amazon.com/MacCarthy-Cross-Examination-Terence/dp/1590318862
I'm not sure if this is exactly what you're looking for, nor do i know about the export options, but found these.
https://play.google.com/store/apps/details?id=com.modelmakertools.simplemindfree
https://play.google.com/store/apps/details?id=com.dripgrind.mindly
There are some good Android apps if you use that platform.
Simplemind or Mindomo are the better ones I've found, but there are a lot to try.
I'm not sure if you're just looking for a simple word search or something more structured. A mind map tool like SimpleMind might be what you're looking for.
This is a free mind mapping website https://www.mindmaps.app/
I don't know if there's any with an app included but I have always used this one https://play.google.com/store/apps/details?id=com.modelmakertools.simplemindfree
Try mind mapping to take this further.
"Mind map. ... A mind map is hierarchical and shows relationships among pieces of the whole. It is often created around a single concept, drawn as an image in the center of a blank page, to which associated representations of ideas such as images, words and parts of words are added."
Here's an Android app that looks interesting. I'm sure iPhone has one similar.
https://play.google.com/store/apps/details?id=com.modelmakertools.simplemindfree
P.S. From my experience as INTJ, you will always be using this type of analytically inquisitive skill just to satisfy your need to understand deeply and come to the best conclusions you can. I find the process itself usually more satisfying than the outcome. You may not always like what you find at the end (like your possible need for validation ;-)).
Simple mind. It also has a desktop version.
SimpleMind is a good one, it's got a few different styles of mindmaps as well.
You could try Simple Mind Map. https://play.google.com/store/apps/details?id=com.modelmakertools.simplemindfree
I like SimpleMind. It has a free (limited version) and paid and has a premium plan, but it works nicely. There's also Mind Meister, but I'm not sure if it syncs across devices.
Free version: https://play.google.com/store/apps/details?id=com.modelmakertools.simplemindfree
Paid version: https://play.google.com/store/apps/details?id=com.modelmakertools.simplemindpro
Website: https://simplemind.eu/