The term "feudalism" often seems to be regarded as a deprecated term on /r/AskHistorians, but what terms are preferred?
Suppose I want to talk about a governmental-military arrangement in which people owe others service in exchange for land tenure, with reciprocal obligations between immediate superiors and inferiors taking precedence over broader obligations (e.g., to a supreme monarch or state). What should I say, in lieu of "feudalism?"
The term(s) needn't be a direct replacement for feudalism. In fact, they'd better not be, or we've actually achieved nothing in understanding in ditching the term feudalism. However, if, for example, feudalism as traditionally conceived actually had several components which aren't very closely related (e.g., a land-owning military class, a lack of strong central authority, etc), and it would be better to consider these separately, what are the terms for those components? Or, alternatively, if feudalism is best thought of as a particular case of some broader, more general pattern (maybe something like a Kautskyian "aristocratic empire"), then what's the terminology for that?
Suppose I want to talk about a governmental-military arrangement in which people owe others service in exchange for land tenure,
In the abstract, as you have phrased it here, "feudalism" is a perfectly lovely word to label the generic style of institutionalized governance you described.
The problem is not the conceptual notion of feudalism as a historical concept. The problem is its blanket application to 1000 years of history across an entire continent. Some parts of the European Middle Ages at some points in time certainly did witness "feudal" vassal-prince relationships like you defined, but always in conjunction with other types of governance relationship--for example, princes developed formal political relationships/rules of homage and exchange with each other as equals. Princes in western (modern day) France often simultaneously held some lands under the French king, some lands under an Iberian king, and perhaps even some of their lands as "vassal" to no king but themselves! The obligation of military service was much more often financial, for the purposes of the overlord acquiring an army.
Again, that is not to say that what we think of as "feudal" governance relationships never existed. However, many medievalists are reluctant to use the term at all, because it has been SO powerful in medieval history. We hear "feudalism" and automatically apply it to homogenize the entire era all over Europe.
When teaching, I usually talk about "versions of medieval lordship" or "types of territorial lordship" to emphasize the diversity, and then talk about what specifics existed in the time or place in question. To go with your examples--I might pick someplace with weak, quasi-non-existent-at-some-points central authority like various points in HRE history; and someplace with a growing central authority like later medieval England.
I, personally, don't have a problem with saying that sometimes these did involve "feudal" relationships--as long as we define clearly what "feudal" means in a specific context. But again, some people think the term is just too tainted and meaningless for medieval Europe that they won't use it at all, and talk only in examples.
I cannot say anything at all about the use of feudalism to describe situations outside of Europe.
Just a note that I've removed the [META] tag from your question. The [META] tag is reserved for discussion of the subreddit itself or its rules. Your question is on a historical subject.