The only person I've seen that I can remember was Phoenixstar who played as them in one of his first games.

Well I disagree goa, there is one thing that they are good at, better than the Rebels at least. They have a pretty good "build order" for economy. Once the Advent-Rebel gets to tier three there is pretty much no economical tech that they can get until Allure. But the loyalist can research assimilated populace(Tier 4, this tech should also be in perception in the Harmony Tree <__< just a pet peeve.), then go into global unity(Tier 5). But that is pretty much all they have going for them right now. 

But thats pretty much the only strategy that I have come up with at the top of my head.

OOps, forgo about expulsion. But its very niche, but I suppose it counts. I do believe all you need to do is take over a planet, it doesn't matter if you conquer it by evicting an enemy first. I don't actually know though because there is no info in the info panel to actually show if it is working or not. And I find global unity very useful because you only need to place one temple early game for the economy, but if you keep expanding that temple will barely catch up to the rate of your fleet. I got out of a game and I expanded to about 5 planets, the culture from the original tower was at the third planet, then I got global unity and continued to expand. 

I'm sure you can use it for military tech however at the same time.

Also I would like some insight Goa, what exactly are the roles of the Advent. It seems like Stardock dropped the ball when they were created. And what I mean is take a look at the other races. I compiled them and assigned them roles to help demonstrate what I mean.

Race--Dominion--Role--Vanilla Role

TEC-R Offense - Expansion - Economy
TEC-L Defense - Production - Economy
Vasari-R Defense - Diplomacy - Mobility
Vasari-L Offense - Economy - Mobility

This is kind of where I get confused with the advent and instead of giving a simple description I'll go a bit more in depth. The Advent rebels are implied to be the more offensive ones, as they cleanse the unity. But they have wail, almost a 100% defensive ability. They also have protection of the unity, and defensive spells that revive a portion of the fleet, lets not forget recharge shields. It seems like the Eradica should of been the Loyalist Titan, with some ability differentiation of course. So in the graph above they are something like

Advent-R Defense - ??? - Culture

The same identity crisis goes to the Advent-Loyalist as well. Alot of there upgrades suggest you be offensive. Fury of the Unity, Assimilated Populace, and Ancient Retribution all suggest offense. But culture innately is very defense oriented. And there titan kind of fits the offensive theme. But far more poorly than the Advent Rebels. They are more clearly defined however than the rebels.

Advent-L Offense - Culture(economy?) - Culture.

I think they just need some new tech and adjustments to the titans. I think with this question answered we can direct buffing the loyalist in the right direction.

Honestly I don't see how they live when the Eradica and Ragnarov fleet wipes them. I don't think they are terrible but they are the worst.