Great work with the chart!

The one job that peaked my interested from the chart was Engineer. To me it would seem to be a combination of Arcanist and one of the Disciples of Hand classes. Perhaps the only requirement would be to have one DoH class leveled to 20. The more 20+ DoH classes you have, the more skills an Engineer has access to.

As for what they would do in battle, I haven't the foggiest. It's not like there is a plethora of mech-based enemies wondering around in Eorzea that they could beat up or take over. Maybe Engineers would be the FFXIV version of a puppeteer.

You call out your machines during a fight, ones which you have built prior. (The opening Engineer quest would be to collect the materials you need to craft your first machine.) Different machines exploit different weakness of an enemy. And, of course, they need to be repaired occasionally, like any other piece of equipment.

Each DoH would have a specific, albeit simple, machine associated with it. While the more complex and complicated machines would require cross disciplines support in order to build and maintain.

The Engineer's motto, "I don't use what I can't build."

And while we're at it, since I didn't see it mentioned anywhere...
Ninja > Thief > Ranger
Bard > Dancer > Conjurer -- no idea what classes could unlock this