Archaeologist of medieval Ethiopia here. As one of the moderators said, this is a rather broad and difficult question to answer. There was certainly some missionary activity at different periods, but a number of factors likely contributed to limiting proselytizing efforts at different periods and places. One thing to keep in mind is that unlike many of the other churches you mention, Ethiopia was part of the Coptic Church in Egypt, which remained a relatively remote and sometimes antagonistic authority toward Ethiopia. The metropolitans sent by Alexandria almost never knew anything about Ethiopia and were mostly figure heads; the real head of the church in practice was often the emperors, whose interests in religion were often as tools of state power. More broadly, constant internal divisions among competing noble states, frequent outside conflicts with Muslim states, and a steady migration after the 1600s of Oromo peoples northward into the highlands often meant that the interests of the state were inward looking rather than expansionist (geography and Ethiopia's "natural barriers" often come up too), trying to maintain unity at home and prevent external incursions. Steven Kaplan has a good book called "The Monastic Holy Man and the Christianization of Early Solomonic Ethiopia." He argues that while the state and its concerns for religion were often mostly inward looking, it was often individual priests or monks who were at the forefront of religious conversion outside the state, and that often state expansion followed on their tail. He's really only dealing with a few centuries from about 1300 to 1500 though, after which Ethiopian society and politics changes radically with the destructive wars with Gran, the arrival of the Jesuits, and the slow decline of central political authority. Energy toward conversion was often put toward keeping the often fractious Ethiopian church in unity (the movement/heresy of Ewostatewos almost began a civil war in the 1400s), keeping Muslims citizens and merchants "in their place" as it ware, and converting invading (a strong word perhaps, migrating might be better, but the Ethiopians very much considered it a boon and a bane based on the circumstances) populations like the Oromo.

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Again, it's a pretty difficult question to answer succinctly as there is no one reason a particular emperor or period of time made a conscious decision not to proselytize. I think in an ideal world, pre-modern Ethiopian society would have loved to convert the rest of the continent as they knew it, but I dont' think their concerns or identity were really tied to it the way, say, Europe was around the period of the Crusades. I recommend you check out a few of the following books if you're really interested; they'll provide you a broader context for the role of the church in Ethiopian society.

Kaplan's book mentioned above.

Tadesse Tamrat (1972) "Church and State in Ethiopia"

Donald Levine, "Wax and Gold"

(if you can read French) Marie-Laure Derat (2003) <<Le domaine des rois ethiopiens>>

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Edit: regarding the particular phrasing of your question: "Why didn't Ethiopia spread Christianity into Africa," it's a bit misleading. Ethiopia has changed much since Ezana's conversion around 330 AD. We don't know the definitive boundaries of the Aksumite state, but Christianity early on was almost certainly a religion of the elites, later sweeping over the lower classes. Christianity almost certainly spreads by way of conversion and adoption for centuries afterward, exemplified by the ethnic-Agaw Dynasty that claims succession to Akum on religious grounds; and the even further south Shewan Solomonic Dynasty that succeeds it. The size of Ethiopia grew substantially after the Aksumite period (though it would contract and re-expand many, many times) and where the state went, Christianity spread as well. The maximum expansion of the state was largely up to its natural geographical boundaries of the Western Highlands and for practical reasons, never really managed to move beyond these borders after the Aksumite period. So, Christian Ethiopians did expand into more of Africa technically, but never really beyond the mountainous region it came occupy. Also, don't forget that during the Aksumite period, most of Sudan, Egypt and the Red Sea region had substantial Judeo-Christian communities before and to a lesser extent after the spread of Islam, so, it's not as though the religion was so unknown toward one important frontier.

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Edit 2: Forgot to mention, J. Spencer Trimingham, "Islam in Ethiopia" (it's really mostly about Ethiopia's foreign relations in the pre-modern period, with religion obviously a focus)

Hey there,

Just to let you know, your question is fine, and we're letting it stand. However, you should be aware that questions framed as 'Why didn't X do Y' relatively often don't get an answer that meets our standards (in our experience as moderators). There are a few reasons for this. Firstly, it often can be difficult to prove the counterfactual: historians know much more about what happened than what might have happened. Secondly, 'why didn't X do Y' questions are sometimes phrased in an ahistorical way. It's worth remembering that people in the past couldn't see into the future, and they generally didn't have all the information we now have about their situations; things that look obvious now didn't necessarily look that way at the time.

If you end up not getting a response after a day or two, consider asking a new question focusing instead on why what happened did happen (rather than why what didn't happen didn't happen) - this kind of question is more likely to get a response in our experience. Hope this helps!