Moreover, I would suggest that the vast majority of people who play L5R do so in order to play magical fantasy samurai (to a given degree of 'magical' and 'fantasy', which changes from player to player), not in order to interrogate the inherent contradictions of the society with regards to the lower classes, and its consequences for ethical standards in the setting.
You can't really have one without running into the other. Unless the players avoid non-samurai like plague. And once someone runs into the problem, the djinn will be out from the bottle... I mean, I did not mind the caste system until my Utaku Battle Maiden decided to adopt an eta girl.
"No, sorry, you can't be an honourable Lion without contesting the Lion traditions of oppressing the heimin, because they fundamentally contradict modern standards of moral behaviour."
The funny thing is that it contradicts in-setting standards of moral behavior too (namely the Bushido tenets of Compassion and Courtesy, and probably Duty and Honor too).
(sorry for the double-post, and the formating; I fail at editing this forum's quote boxes)
To address your first point: you don't have to ignore them 'like the plague'. You just have to have no meaningful interactions, which is easy enough, and in my experience of the RPG, fairly normal, outside campaigns that are deliberately intending to raise these issues. Sure, heimin are responsible for making sure virtually every part of a samurai's life runs smoothly, but that doesn't mean the samurai has to pay them any attention. Since you mention an Utaku character - the Unicorn are pretty weird in Rokugan for taking a personal interest in their heimin. The idea of adopting an eta child would be literally unthinkable for any Lion or Crane, and if it never crosses your mind in the first place, you have little reason to wonder why it should be so.
To your second point: it's a truism that the Lion are bad at Compassion, but there's no reason the Lion themselves should see it that way. To quote Leadership :
"Through intense training the Samurai becomes quick and strong. He is not as other men. He develops a power that must be used for the good of all. He has compassion. He helps his fellow men at every opportunity. If an opportunity does not arise, he goes out of his way to find one."
Notably, that quote doesn't say "don't be a jerk". It says, "use your strength on behalf of others, and help people as much as possible". Even setting aside the question of whether or not heimin qualify as 'fellow men' for the purposes of Akodo's guidance (they don't really, in the opinion of the Lion), the Lion could quite plausibly see no contradiction between those instructions and their own traditions, as long as they did something nice for the people- a condition which, in the view of most Lion, is perfectly adequately served by their maintenance of security, and the traditions of the Empire.
As for Courtesy, again, heimin aren't really people. If anything, treating peasants politely would be considered an insult to everyone else, as it implies that samurai are worthy of no more respect than heimin- think of the outrage among veterans who had received the Légion d'Honneur, when the French announced that they were awarding it to a (particularly heroic and life-saving) carrier pigeon.
Finally, the Lion generally believe that heimin were born as heimin because of their actions in past lives, and that the best way to help them to achieve a better rebirth next time is to ensure that they fulfil their duties as part of the Celestial Order properly- a goal that is best served, in the opinion of the Lion, by constantly reminding them of their status, and not allowing them to get any delusions of grandeur that might cause them to act in ways inappropriate to their position. It's undeniably nasty, but it's perfectly logical within the setting's cosmology, and allows the Lion to maintain a clean conscience.
In short, Bushido is a code for the maintenance of social order- it's all about obligations to others, both horizontal and vertical. It is not about 'being a good person', as we would understand those words.
To be clear, I'm certainly not defending the in-setting caste system. It's oppressive and reprehensible, and absolutely should be repulsive to us. It is perfectly reasonable to have doubts about the ethical status of those who uphold it most strictly- it would be more worrying if there were no such doubts. If the setting were to be changed so as to be more in line with modern standards, the caste system should absolutely be at or near the top of the list of things to be drastically changed. Within the setting as it stands, however, there's no reason it should come up as an issue with any frequency, nor would it be fair to deny the Lion (and Crane) their status as shining beacons of wonderfulness within the setting because of it.