The question was "ancient" and "organization". The Mongols were tactically excellent, and for a brief period of time swept all before them.
To say that at period of conquest they were just
tactically excellent, is completely wrong. Their campaigns in the CA and Europe had been samples of the
strategic excellence unsurpassed until the modern times. Coordination of the armies advancing on a front of few hundred miles with an ability to get messages from the troops 300+ km away within 3 days was something close to impossible before well into the XIX century (Nappy could not arrange for the reliable communications within tens of miles and coordination of the corps actions was something of a systematic problem).
Their "organization" was basically tribal
Oops... The major difference between the system introduced by Genghis and the earlier nomadic system was in abandoning formation of the troops by a tribal principle: it was retained only on the low level formations (up to hundred).
and based on personal leadership and loyalty.
Starting from Marius armies of the Roman Republic had been recruited based pretty much on the principle but it seems that you are more than a little bit confused about application of that principle in the Mongolian armies of Genghis. Of course, loyalty to the supreme ruler (Great Khan, emperor) was a paramount for having and effective army but as far as leadership on the lower levels was concerned, the Mongolian top commanders did not have their own armies. For example, Subotai (the best-known figure) had been more than once transferred from one theater to another and the armies of which he was de facto commander during the European campaigns were not
his armies or the armies of a formal commander, Batu. When, after European campaigns, he was transferred to China, he did not come there with his personal army (which he did not have to start with): it was just an individual transfer of a highly respected military specialist to a new position.
As light cavalry their logistic needs were relatively light, however they needed to move because of the need for grazing for their horses, a supply of fodder was not part of the system.
The Mongols were not exactly a "light cavalry": they had plenty of contingents heavily armed and armored by the standards of time. And an idea that they did not need a logistic is quite interesting: are you implying that the soldiers also had been eating grass?
Or that they did not have any luggage, did not need to arrange for replacement of the horses, did not need new arrows, and the list is going on.
Multiple remounts per warrior, but a "logistics chain", no. To the extent they had combat engineers they were subject peoples. Again, this was not a knock on their combat effectiveness but an issue of "organization", compared with the Romans with a training system to create disciplined soldiers, the construction of fortified camps on the march, forts (castra) to a more or less standard plan, support functions including a central medical facility.
This probably implies that the Mongols did not have any training needed for fighting in a formation and accomplishing rather complicated maneuvers which would surely make them into some super-soldiers with that ability magically being extended to the subdued people added to their armies. Actually, they had been conducting extensive training on all levels and even conducted what in modern terms would pass for the army-level maneuvers (regular mass hunting involving maneuvers by the army units). Of course, they did not have the Roman training system but Mongolian discipline was at least as strong as Roman. Their military organization was well documented and quite complicated and logistics had been a part of it. While they were not building the fortified camps, they had quite effective organization for making the camps with a high ranking officer responsible for the arrangements. It was quite different from the Roman system but this does not mean that it did not exist or was inefficient.
One measure of organization is staying power, the Mongol Empire broke in to pieces rather rapidly, compared to the substantially longer staying power of the Romans. Furthermore the Romans had a system for veterans, giving them land grants for extension of Roman power. While various promotion, awards, and larger grants were based on performance, every soldier who completed his time satisfactorily received his "pension".
The reasons for the Mongolian Empire falling apart relatively fast but an absence of the pensions was not one of them. You can start with the system of appanages, which made its disintegration inevitable. Then, the very size and cultural diversity of their empire in which they were a tiny minority and the least "cultured" one made the task of keeping it together impossible even under the most favorable circumstance: how to make China, Central Asia, Middle East, Russia into a reasonably uniform cultural entity? Especially if your "core nation" is something like 1 - 2 % of a total population? The Romans never had empire of a comparable size and never had been at that demographic and cultural disadvantage.
I would venture to say that a Roman legion or two could not stand against a roughly equivalent number of combatants of a Crusader army. Longbows, crossbows, and the charge of armored knights would do the trick.
I wonder which "Crusader Army" it would be, taking into an account that Richard had crossbowmen, not longbowmen, and the later crusading armies did not have significant numbers of the English. Charge of the knights against a high quality infantry, especially if it manages to put on even minimal field defenses, would be a dicey thing at best as had been demonstrated more than once against the infantries that were no match to the Romans in the terms of organization and training. Let me see, Courtrai, Nicopol, Mohacz, etc.