Why did the Ascians never try to do these?

[Veloran]

To be fair, his diplomatic approach was not in good faith. It was another calculated scheme. He bought himself time to course-correct and learned more about what was going on with his enemies (as the Exarch had just done something impossible).

His diplomacy was rooted in pragmatism. There was a small chance the Warrior of Light might accept Emet's perspective that they "only stand to gain" from the Rejoinining, and then, if not, there was still a larger chance that the Warrior of Light could be used to control the Light until they eventually transformed into the very monster that would usher in the Calamity course-correction. And if neither of those happened, he could always just kill them. It was win/win/win. Until it wasn't.

This is probably not the case.
First, due to the nature of the rift, Emet-Selch effectively had all the time in the world to "course-correct". To wit, the first time chronologically we see Emet-Selch is actually in WoL's Echo vision of Vauthry, where he approaches Vauthry's father. This was some time after G'raha arrived on the First with the Crystal Tower, but decades before he would attempt his first summoning and end up with Thancred. Assuming any linearity to time, the first time we see Emet on the Source he's already been to the First, saw how the forces of the Crystarium and Eulmore were trying to combat the Sin Eater threat, and had taken action to counteract them by disrupting their alliance. Considering this, he had decades of time on the First to deal with the issue of the Crystal Tower and how it had clearly traveled through time and the rift, but he chose not to do so.

As he says, Emet-Selch could have simply sided with Vauthry outright and killed everyone. But as we see in his short story, he has a tendency for "hope against hope". Rather than being aghast that "Azem" either was WoL at all or that they wouldn't join him, I would say his reaction to seeing Azem in WoL and saying "No, it can't be.", is pure disbelief that his vague hopes for some kind of alternative solution might possibly bear fruit. We know Ardbert is directly paralleled with Emet in how he had lost hope over the course of 100 years, and that it was unimaginably worse for Emet over the last 12,000 years, so in the same way Ardbert had to be pushed to find new hope in WoL, Emet can't even believe his eyes when seeing it for himself, despite wishing it to be so.

And we can certainly say he did want it to be so, considering he prepared all of Amaurot and the plan with the Azem stone and seemingly preformed some kind of Ancient funerary rite in revealing his true name before the final battle.

Basically he got the win he'd always wanted but didn't think was actually possible, until it was.