You're mistaking modern-day Ethiopia's borders for that of historical Ethiopia. Historic Ethiopia/Abyssinia traditionally consisted of the northern regions of the country today - the Tigray, Gondar, and Wello regions, along with the northern part of the Shewa region. At least by the latter half of the 19th Century, Ethiopia very much was attempting to project its power upon its neighbors, but these neighbors were a lot closer than you may realize. Ethiopia fought Ottoman Egypt, which had extended its influence down to Sudan and the modern-day Eritrean coast, later having to fend off the Madhists who took power in Sudan. Successive emperors did attempt to push Ethiopia's borders south into regions not traditionally controlled by Ethiopia (Oromo-dominated southern Ethiopia and Somali-dominated eastern Ethiopia), but constant warfare with both enemies abroad and rebelling nobles at home meant that projecting power even to weaker neighbors was difficult.

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We only really see modern Ethiopia's borders take shape following the First Italo-Ethiopian War, where Ethiopia's borders were defined as more or less what they are today. For then-emperor Menelik, this was a major victory, as it not only had secured Ethiopia's sovereignty in the eyes of the Great Powers, but greatly expanded Ethiopia's territory from its core in the central and northwest regions to include Oromia and Ogaden. By that point, however, it was 1896, and Central Africa was now firmly in the hands of the British.