I think there is one major point to consider: training. How many Mercs who work as infantry / look for outfits that hire infantry are actually trained in using the equipment? I get that the Merc unit might train them in that but let's say fresh on the market soldiers? I somehow doubt that any house lets this specialized infantry leave. Unlike leg or motorized infantry I think Jump-Infantry requires more training and therefore is too valuable too simply let go. Plus I am not sure that especially in the rather tech starved days those soldiers can actually simply go and thake this kind of equipment with them (Soldiers who go AWOL / desert (get left behind might be different)
And perhaps another aspect: Jump-Infantry are often used as shock troops and have most likely high casualty rates. While the Great Houses can replace / rebuild such units with more troopers I feel that Mercs might have more trouble rebuilding Jump-Infantry units and therefore might stick to more conventional infantry. Of course big units with their deeper coffers and perhaps even own training units (You meant the Illician Lancers) might have less problems at that
By that logic the Houses would never let mechwarriors and aerospace pilots leave. Perhaps the same with dropship and jumpship crews. As they spend years training those, typically expensive academy level training. But they do go, and they do become mercs. So that logic doesn't hold up. Jump infantrymen for all their expense, don't hold a candle to those other military professions in terms of cost and investment and they leave all the time. Also, it's hard to MAKE someone stay if they truly want to leave and have completed all of their military service requirements and obligations.
But I get your point on the casualty rates, and that's probably the best argument I've seen that I can understand and agree with. The relative cost of a human being, in a Great House military, with a Great House that intakes thousands of soldiers every year, is probably very different math than it is for a merc unit. And of the infantry unit types, jump infantry are probably among the fastest to get chopped to pieces by intense combat action in the kinds of situations they get dropped into. A Great House military is going to be quicker to react with a sigh and a "it is what it is, and in the name of our nation and our Great House Leader, it is necessary" to that reality, while a merc unit might decide they just can't stomach it.
That argument I get, and it does jive with the canon-statement from FM: Mercs as to how/why mercs are unable to accept the casualties that occur when infantry are employed in front-line combat.
Even then, even with that, we have seen infantry-centric merc units, and so it still surprises me that at least one merc unit didn't pop up centered on this niche. In a universe that has units like Kraken Unleashed, and Stalwart Support, and Thor's Hammers, an artillery-centric unit (again very niche units). I'm genuinely surprised that no jump infantry battalion/regiment centric merc unit has popped up. I think the Great Houses would love to be able to hire on additional jump infantry. Considering their rarity, difficulty to train and so on.
It might take a Wild Geese-esque situation (House unit masquerading as mercs) to make it happen, to have a unit that can justify the human casualties and sustain them with some House-supporting infrastructure and a House-supported pipeline of personnel or training infrastructure, but I'm honestly surprised it never did happen. But with the advent of battle armor, I can see that the appropriate era(s) for such a thing are now historical to the BT-universe.