Tytalus was a theurgist; he made pacts with the Titans for the power necessary to kill his teacher Goerna. Presumably the Tatanoi are about gaining that kind of power to deal with those that oppose them. Looked at critically, the Titanoi need no other goal. Many titans will have high Magic Might and thus require powerful rituals with great Penetration to conjure, so having assistance in the summoning (via Wizard's Communion) is absolutely vital. So the need for an organization of like minded theurgists is there - you initiate other magi so that they can help you summon the big stuff in return.
As for diabolism, if you're already a ReVi specialist, controlling demons is awfully easy...
And I don't subscribe to the view that Titanoi are involved in any actual religious observance related to the Titans. HoH:Societates, page 71 discusses Guorna and her magical practice. She used the gods of the dead "(despite the secularization of their religion)."
Guorna may have viewed her work with religious underpinnings, but I don't get that sense from Tytalus (and by proximity) Tremere.
I'm going to go with the suggestion that the magi of House Tytalus c. 790-800 are still fundamentally driven by the same goals as the canonical ArM5/HoH:S Tytalus magi: growth & learning through conflict. That is, they will NOT be as unified as most Mystery Cults are; instead, most individual magi will have their own individual agendas which they rank above the Cult agenda(s), and they will act accordingly. A wholly-corrupted/infernal magus will have goals entirely in-line with some Infernal agenda, of course;
The pre-schism outlook is liable to grander visions, more arrogance, less caution/inhibition, etc; but still growth-by-way-of-conflict, and plenty of room for individualism.
I think of the Titanoi as selfish magi that summon great Magical spirits to do their bidding, rather than worshipers or a religion per se. They don't obey or respect the spirits, they just use them as a source of power.
As said above, the need for high penetration keeps the society functioning, somehow. And they're ripe for Infernal corruption, with anything from Corrupted Arts to Binding.
I don't see the schism as making a difference in any of this. The Betrayal/Corruption does, by making them more cautious and perhaps less corrupt. But even that's not such a major change.
If we think than the Titanoi are paying favors, more than they were loyal whorshippers or servants to the Kosmocrators/Titans, then i put that really they are not so easy allies to them. And I think that not all Tytallus are Titanoi, more of them won't see him like servants of other beings, they could empathize with the Titans against the Gods, or they could understand them, but not serve them.
I think another common goal would be to push back the faerie realms as well as the Divine, and having mortal servants of the magical realm would possibly be useful with that. But beings as powerful as them could also have multiple facets that mortals couldn't understand. Titanoi and jotuns/titans might be the same thing but mortals can't conceive of them that way.
For my purposes i would have the Titanoi and Jotuns/Titans be close enough to have similar purposes and goals though certainly not a big happy family.
I'd see it as a "we will ally with that other group, but at the first sign of trouble we will betray them, as their long-term goals make them inherently untrustworthy." - that stance being used by both sides, of course.
Less on the Tytalus side, though - mainly due to the calming effects of Parma, more than anything else.
In my current (probably defunct, sadly) saga, the Titanoi and Muspelli are actually enemies. The Titanoi are all about wielding buffed-up Hermetic Theurgy to command the lesser titans and channel their power against their will, and serve as the Bad Guys for the saga. The Muspelli are all about worshiping the titans and doing their bidding, and serve as their liaisons to the Order against the Muspelli, connecting the PCs to a pro-Titans cult (a version of the Huntress in the Wood) and then directing the cult and PCs against the Titanoi and to further the titan's release.