The "house elf problem" does not, to my knowledge, yet have a name. It is a good distillation of a general problem with individualistic/rights-based ethics.
The general issue is based on the reality that individuals are not the only, or often even the best, referent that one should use for either moral or other optimizing framing. If, for instance, one is interested in species preservation, then the welfare of a single post-breeding individual of the species should not be a concern. If one is interested in the health of Gaia, the biosphere, then even the continuation of an entire species would be of minimal concern, because Gaia operates with regular species replacement.
For we humans, the welfare of a household, an extended family, or a small community is often of far more importance than the welfare of each individual member of a community. Contemporary western moral thinkers operate within a massively successful community, where there is significant survival margin and therefore a high tolerance margin to address secondary concerns such as freedoms and self-actualization. If we lived as most humans have in history -- one the edge of starvation, then many of the intrinsic prejudices we humans have which are problematic for our more modern and comfort-based moral sensibilities, would make much more sense.
For instance, we humans are somewhat eusocial -- we support the others within our community in times of hardship. This is a moral virtue, and increases our ability to individually survive erratic fates, but it imposes a carrying cost on the community for everyone who is seen as "us". Strangers, foreigners, whose commitment to supporting "us" themselves when the rest have hardship is unclear, SHOULD be regarded with suspicion when resources are scarce. Even more brutal, those who are disabled from birth, "cripples", were a burden that communities often could NOT afford to support for a lifetime -- and infanticide through exposure was often a justified community response to reduce the risk of wholesale starvation due to privation.
In this context, that social advantage sometimes trumps individual disadvantage, we can discuss the consequences of the reality that humans generally have to organize hierarchically to do things done that require more than a very few of us to get done. One of the consequences of humans having innate tendencies to organize hierarchically, is that we on occasion SEEK strong leaders, rather than always pursuing individual self-actualization. Followership is built into us as part of our evolutionary makeup. All of us have some degree of house-elf in us.
We can see this more clearly in creatures that are more communal than people are. Horses, for instance, are happiest when they are in a herd, with a strong domineering herd leader. This is something that horse trainers can take advantage of -- making THEMSELVES the herd leader for the horse, assists tremendously in training a horse for useful/safe riding. Dog training uses similar principles, a well-trained dog recognizes their people are the pack leaders.
House elves take these principles to a logical extreme -- the HOUSEHOLD becomes the unit of value, the humans are the heads of the hierarchy, and the house elves ARE happiest when purely subservient. This is different from dogs and humans. For us and dogs, each of us may at some point need to serve as leader, hence we each have both leadership and followership traits. Plus dogs and humans also operate outside the household/pack, so must be able to operate competently autonomously/equally as well as hierarchically/communally. Making subservience a species trait, not just a temporary role -- well we humans did that to wolves to make dogs...
At any rate, hierarchies, subservience, followership, and self-effacement are crucial traits for communities and organizations to thrive. They are "problems" for purely individualistic ethics. But if one can change one's framing, adapting one's reference system to different scales: from genes, to cells, to individuals, to communities, to species, to cultures, to ecosystems, or to Gaia, as appropriate -- then followership and "menial" roles are no longer a problem, but a benefit.