She exists as the only (known) antithesis to the Solari people, the only moon worshipper. If the Solari were distressed enough by her alternative thinking to schedule an execution, why haven't they dispatched an assassin or a garrison of troops to kill her.
Probably because the Solari have no history of using assassins that we've seen, at least going back to the time of the original conflict, have no real reason to have any sort of standing army with the Rakkor as their acting protectors and lost a huge part of their leadership structure to Diana's ascension. Leona went from being something of a figurehead to the solar cult to being very close to the top in terms of influence and her conflict with Diana is a very personal one. She isn't detached enough to send someone else to do the job, and it's hardly as if the heretic has a known location they could send any force meant to deal with her anyway.
Likewise, why hasn't Diana become a preacher of sorts? I know she's vengeful and wants to witness the fall of the Solari, so why hasn't she accumulated a massive following and marched them to the gates with rage in their hearts?
Probably because, just like Leona, Diana considers this a very personal conflict and sees it as her own duty to 'enlighten' the Solari - or kill them. Rioters have said previously in AMAs that the Scorn of the Moon is so focussed on her agenda that she hasn't even given thought to how she would go about rebuilding her order and indeed doesn't even have a real plan for what she would do should she "Bring down the sun." She isn't a cult leader or some imposing orator, she's a person who saw a 'truth' very personal to the Solari and wants to break them just as they tried to break her. Diana's goal is to show the Solari how wrong they are and to remove their evil from the world, not to enlighten those beyond the conflict and use them to fight her battles.
She should link with Elise
and her followers, and seek to usher in an era of eternal darkness, where the moon reigns supreme and Elise and her spiders rule the night and feast on the captive Solari.
What on earth would Diana want to do with Elise? How would she even meet her in the new canon?
Elise is pretty much everything Diana hates in a religious figure: Someone who abuses the trust and loyalty of others for personal gain, twisting a 'god' from something that exists only to feed on its followers to some kind of benevolent deity worthy of tribute.
Expanding on Mt. Targon, we don't know ANYTHING about it, or about the hidden temple where Diana found her suit/crescent sword and the people who built it/worshipped in it. Were there once two separate people? the Solari and this unknown moon-based group? Were they always enemies or did their relationship sour?
We don't really need a massive change in the nature of the conflict for this.
Additionally, Diana could request the aid of Noxus and bring yet another territory into their grasp. This in turn would attract Demacia and provide more conflict for the warring nations. (ALSO MORE SKINS)
Uh ... from a storyline perspective, why would you want to turn Diana and Leona's extremely personal conflict into yet another vehicle for Demacia and Noxus butting heads?
I'm totally in favour of more content pushing forward the conflict between the Solari and their fallen counterpart, but surely any future development should be centred on the two characters who embody those two factions rather than looking to dilute things further and further by bringing in characters who have no real connection to the original conflict beyond "Diana/Leona asked me to help out."