For a Crusader expedition, the Euphrates one, at the time being, is the only one that is reasonably feasible. It is geographically closer and they still have the western parts of Edessa to depend upon. The Tigris is too far deep into hostile territory, and, as you mentioned, there won't be a solid logistical base to allow for supplies.
I should have precised further political motivations for either one.
As the Euphrates route is feasible as soon as Edessa is secured, it's also the route that will be available very soon. And this is a problem, for the Basileus, as it draws the crusaders away from Armenian highlands which don't need to be secured for this route to be used.
Other advantages for this route to precise, any crusader army here would have its back against the desert, so doesn't have to worry about being surrounded; at any danger, provided riverine support is adapted and rearguard action is firm enough, the army can just cross the river to reach safety.
The Tigris route on the other hand plays more into the Byzantines' hands. If the basileus can convinces the Franks to go that way, at least he has a valid pretext to entice them into conquering Armenian highlands first.
Your assessment about France is also fascinating, and you present a very solid case against a possible independent "Kingdom of Aquitaine". Once again, there is a lot of ground to cover before we get there, so I really won't be making an effort, right now, to dispute your points. On the contrary, I believe you might concede me the possibility of revising some premises later on; perhaps indeed we won't be seeing.
When I brought up the idea of an independent Aquitaine, it is a very embryonic thought, I really did not think (yet) seriously about the details. For the time being, my only premise is: WI Eleanor's patrimony continued with the House of Poitiers instead of merging into the French Crown? Then, we can discuss new possibilities. In any case, it certainly won't happen before the 13th Century.
Now, I'm not really considering a Balkanized or HRE-like France, but rather a movement in which the Capetian centralization is much less successful than IOTL. It was far from inevitable, and took the likes of a Phillip II Augustus to happen as it did. If we see the vassals keeping their own shares of power, be them allies or not to the Crown, we can expect, in counterpoint, that the French monarchy remains a fairly weak one, but with oscilating political fortunes. I'm thinking less about the HRE and more about Poland-Lithuania, meaning that strong monarchs made the monarchy much stronger, but, in the long scheme of things, there was a trend for provincialism and aristocratic decentralization.
My bad if I misunderstood your original intent on Aquitaine, but it's good we set it clear now.
And it goes without proverbially saying it that I do concede ^^.
As for the centralization, I'd say you can make it way slower than less successfull.
The Polish-Lithuanian example is not very relevant here, and for quite big reasons.
As I said, the Kingdom of Franks (effectively, OTL, Philip II went from Rex Francorum to Rex Franciae) was not really anything near the elective monarchy that the Commonwealth was, and there were no such thing as liberum veto or sejm. Political decisioning, even embroiled amidst feudal conflicts, was much more effective and decisive.
Then, as the dynasty had an exceptional longevity and stability, compared to feudal lords and other European dynasties (only the Capetian to Valois and the Valois to Bourbon breaks to account for in over 8 centuries), it stood to expand at each passing generation, by marriage and acquisitions at the very least, reversion of appanages (the appanages were granted on the condition of being returned to the Crown in case of extinction of the direct male line) and ultimately, lands forfeited by felonous vassals and ones conquered. For sure, IOTL, the Capetians had great success at expanding through seizing forfeit lands from their English vassals.
So, ITTL, unless you get rid of the Capetian dynasty alltogether, you are only getting to slow down the centralization trend.
To mention, avoiding a conflict in the likes of the Hundred Years War and you will for sure delay that trend for a long time. The continued wars and the financial strain they put on France did much to transform the fiscal and military structures of France, decisively driving the center of power to the King's persona as we would see from Louis XI, the Spider King, onwards. Before that, the monarchy had been very reliant on great nobility, the Burgundians, the Armagnac, the Orléans, the dukes of Britanny, ...
To return on the subject of the Albigensian crusades that has been mentionned above while speaking of the Montforts, I'd say you're right to assume they are butterflied, but not for the reasons you said and believe I think.
IOTL, after the reign of Phillip II, the Angevin threat had been neutered as a result of John I's disastrous rule and Henry III's minority. ITTL, it's implied the status quo remains. England remains solidly anchored in Normandy and Aquitaine is still a powerful vassal. That means that, ITTL, the French King needs Toulouse as a firm ally against Aquitaine and therefore will probably shield it from any papal action, which means the Counts of Toulouse can do whatever they want about the Cathars, ie no Albigensian crusade to happen.
In turn, I'm a little excited by what this means in terms of cultural developments in the South.
Not only the Cathars can continue to practice and thrive in the lands of Occitan, but the overall cultural and religious tolerance practiced by Languedoc lords will stay. I mean that the imposition of royal authority IOTL meant the tolerance enjoyed by the Cathars and the Jews alike went away.
With the example of Montpellier in the 12th century, you can see what I mean.
Sitting as the entrance door of the French kingdom to Mediterranean trade and its cosmopolitan influences, as much Levantine (ties with Provence-Toulouse lords in Palestine and Syria), Byzantine (note here the OTL marriage of Eudokia Komnene to Guilhem VIII of Montpellier), or Islamic ones (especially the influences from Muslim Spain)...
Speaking of Komnene princesses to marry, any plan to marry Komnene princesses to Frankish rulers in the Levant yet ?
I believe I made the case there were quite a few Komnene princesses around, though I should have precised that diplomatically, it's rather an advantage for prospective marital alliances.
Is Prince-Duke-Count Roger married yet? I understood he was still young.