So, to think about an answer to this question, we kind of have to think about what exactly we mean by "legitimacy and respect." Because you're certainly right: western European crusaders tend to always be sort of suspicious of Byzantium, and especially of the emperor, even as they can at the same time (sometimes) admire its antiquity and elegant culture. Where is this disconnect coming from? And where do we see it most?

Constantinople was a major stop on pilgrimages to Jerusalem for most of the Middle Ages. Pilgrim's accounts of the city are usually very impressed: they wonder at its beautiful churches, its orthodox theology, and certainly its rare (and costly) relics and artworks. At least through the early 12th c. Byzantium is still theoretically aware of and interested in what it sees as lost territory in the West. This interest certainly uneven -- the Franks/Holy Roman Empire (sometimes political allies, sometimes antagonists) and Venice (increasingly-important naval allies and theoretical imperial subjects) get much more attention than, idk, the Netherlands or England -- but it's definitely there; Constantine VII, in the 10th-c. De Administrando Imperio ("How to Run the Empire"), includes huge sections on parts of Europe that hadn't been solidly part of the empire since Late Antiquity, some of which is even fairly accurate. There had been occasional theological disagreements between East and West, but I'd say even by the First Crusade (i.e. including the Schism of 1054) these had never really been big enough to cause a problem.

At least in my experience, any "disrespect" towards Byzantium is most explicitly found, at least in the beginning, in political contexts. That is, Latin authors (I'm thinking especially during the First Crusade) usually at least implicitly characterize the Byzantine emperor as not really an emperor at all. The Gesta Francorum explicitly calls Alexios I rex, "king," instead of imperator, but the author is also a huge supporter of Bohemond, whose father, Robert Guiscard, had just fought a war against Alexios. Part of this might just be literary precedent -- I mean, the decadent, effeminate Greek is one of the earliest tropes in Latin literature -- but there's also very much a real stereotype of Byzantines as weak, duplicitous, even cowardly. I'm pretty sure westerners were aware of Byzantium's direct connection to antiquity. They certainly knew Justinian, at any rate, and once the Crusades really get going they also pick up on Heraclius as a supposed first crusader. But many areas of western Europe could claim what, to them, was an equally glamorous ancestry, whether through mythical/semi-mythical heroes (e.g. Walter of Aquitaine, Brutus of Troy, even Charlemagne after a certain point) or honors directly bestowed by Byzantium itself (e.g. the Duke of Venice). After a certain point, and certainly by the time of the Crusades, Byzantium's connection to "ancient Rome" was increasingly insufficient to command the same kind of reputation it had enjoyed in earlier centuries.

(Not to say that the Byzantines don't engage in the same kind of behavior. At least by the time of Alexios, this sort of disrespect certainly goes both ways -- I'm thinking mostly of Bohemond's portrayal in the Alexiad, written by Alexios' daughter Anna, where most of her descriptions of him and of the Normans in general are mostly just backhanded compliments. Towards the end of the Komnenian dynasty, and indeed up until 1204, there will be occasional bouts of violence against westerners. In 1182, Andronikos I overthrows the Latin empress regent and allows an enormous massacre of the Latin inhabitants of Constantinople. But even just before the Fourth Crusade, I don't think it's really clear that the Byzantines saw westerners as a serious threat. That's part of what makes 1204 so traumatic: even though we disagreed with them, we thought we were theoretically on the same side. But the Fourth Crusade seriously complicates relations with the West.)

As time goes on, the political motivations for this antagonism are also bolstered with religious ones. Around the year 1100 we start seeing multiple Byzantine treatises "On the Errors of the Latins," usually about the Filioque or practices regarding the Eucharist. These can be a bit esoteric but they were a huge issues at the time: the Filioque (i.e. does the Holy Spirit proceed from the Father, or the Father and the Son?) is one of the major reasons Byzantium is so slow to attempt reconciliation with the West even as it's increasingly threatened by Turks in the 14 and 15th c. The First Council of Constantinople (381) had theoretically established the equal privilege of Rome and Constantinople, although this is certainly challenged (from both sides) throughout the Byzantine period; patriarchs of Constantinople are usually "officially" identified (on seals, in correspondence, etc.) as "So-and-so, Bishop of Constantinople-New Rome." How much of this ended up being lip service? It's hard to say. There are certainly patriarchs who can tangle with the pope, or at least who think they can, though the patriarch I'm thinking of (Ignatios I) is pretty cantankerous and might just be an unusual case. Crusaders, in any event, aren't answering to the patriarch. The religious element to Byzantine-western friction is certainly present at the beginning of the Crusades, though it won't be a major one until around the Fourth Crusade. I work on Byzantium, so I agree with you that it's a pretty puzzling situation -- but I don't know if we can get a more concrete answer than that.

For further reading on this subject, I recommend Krinjie Ciggaar's Western Travellers to Constantinople: The West and Byzantium, 962-1204 (New York: Brill, 1996) and Jonathan Harris' Byzantium and the Crusades (2nd ed., London: Bloomsbury, 2014). There's also a nice volume from Dumbarton Oaks, The Crusades from the Perspective of Byzantium and the Muslim World, ed. Angeliki Laiou and Roy Parviz Mottahedeh, available online. Let me know if any of that helped?