Is there any reason why medieval soldiers could not have been organized into the order of battle/organization (in the modern sense) of the Roman Empire by a ruler or empire that chose to do so? Integrating crossbowmen, longbowmen, spearmen, light cavalry, heavy cavalry (Knights), etc.
I know the Komnenian era had a relatively organized army in the Eastern Roman Empire, but they were not at the scale of the professional standing armies of the Roman Empire from my understanding nor were they organized into an order of battle as organized as that of the Roman legions. Perhaps an expert on the subject matter could explain if the Komenian era Eastern Roman Army was as organized as the Roman legions or not.
Is there a reason that the Legion order of battle model could not be applied to medieval warfare though adjusted for more cavalary? The tactics and strategy would of course change. The table of organization would also likely change from the structure of the Roman Legion as a medieval Legion would integrate more heavy cavalry, crossbowmen, longbowmen, etc.
I'm not sure I agree entirely with the premise of your question or your definition of "order of battle" so I'm not sure what question I'm trying to answer here. Let's clear up the "order of battle" question first. The way you speak of order of battle it sounds like you're refering to the composition of an army, in medieval terms, the order of battle is the position different units take on the battlefield relative to the commander. Replicating the roman order of battle (in the medieval sense of the term, that is, where to deploy which units) would make little sense since, as you say, tactics and strategy had changed.
Deploy an army in the "roman style" (whatever that is, it's not like the roman armies or tactics were particularly uniform over the course of the roman era) on a medieval battle would not be wise because you have very different army compositions, quality of troops and enemies. So, I guess what you mean is "Why didn't medieval commanders raise armeis with a composition more akin to the roman one?" to which my answer is "what?".
The composition of roman armies changed drastically over the course of the roman era and even varied based on where and who they were expecting to fight.
So, there is no "roman order of battle" as such, nor is there a "roman army composition/structure" that remains particularly uniform over any serious amount of time. That is a problem in itself but even if we were to take a particular part of the roman era, say the roman legions post Marian reforms in a particular campaign, we'd run into the next problem right away, what the everloving fuck is the "medieval" period? It's such a loose term that the length varies depending on region and historian and it is also a time where we see huge leaps in military technology. There is absolutely not uniformity of unit composition or tactics over time or geography.
Now, if we overlook all of this for a moment and answer the question I sort of guess you're trying to ask, "Why didn't medieval commanders field roman style armies" the answer is very simple. Big standardized armies require big centralized governments. The medieval era, particularly the early medieval era in north-werstern Europe, is characterized by decentralization. Every feudal lord would come at the call of his suzerain lord and bring with him his own martial skills as well as those of his retinue, militia, feudal levies and whatever else the terms of his agreement with his lord dictates.
The generally tiered structure of european nobility means that when someone high up calls, he does not call on everyone, he calls on those one tier below who in turn call on those another tier below and so on. When a big army like this gathers, the King or whoever issued the call might not have a detailed possibility to dictatet what troops are brought to the field. He his left with whatever troops his vassals have brought and what mercenaries he can find and afford.
That said, it doesn't mean that there was no integration of troops. A big medieval army would absolutely contain most if not all of the elements you mention and more. However, due to the relatively static nature of particularly early medieval warfare we see few cases where these armies meet in pitched battle. Medieval warfar is a LOT of marching, maneuvering for position and besieging the great innovation of medieval warfare, castles (and fortified towns).
When you're conducting this kind of warfare the quality of your troops matter relatively little. You're going to spend a lot of time walking and sitting outside the walls of someplace. Probably due in no small part to the influence of De re militari (Basically the Art of War for medieval Europe) commanders rarely fought evenly matched battles. Like the Art of War, De re militari stresses that pitched battles should only be fought if you're absolutely sure of victory or if you have absolutely no other choice. It's not a bad strategy.
TL;DR -
If you want more material on medieval warfare I like Medieval Warfare: Theory and Practice of War in Europe, 300–1500 by Helen Nicholson.