During the reign of Augustus, the Ethiopians invaded Egypt and where then chased through the Sudan by the Romans and defeated at their capitol. As far as I know, the Romans just left afterward. I know Teutoburg and complications in the East sort of put halt to the Empire's expansion at this time, but wouldn't Ethiopia be quite desirable for a multitude of reasons- one of which being Augustus' interest in sea routes to India?
During the reign of Augustus, the Ethiopians invaded Egypt and where then chased through the Sudan by the Romans and defeated at their capitol.
This part needs to be clarified. The raid on Egypt was conducted by a Kushite army under the Kandake Amanirenas. The capital of the Kushite kingdom was Meroe, which is situated on the Nile river, north of Khartoum in what is now the Republic of Sudan.
It was a habit of Roman authors to call the people of Kush/Nubia "Ethiopians" and the people of Aksum "Indians"^1. However, modern scholars would not call the people of Kush "Ethiopian", and Aksum is seen by many historians as a predecessor to the medieval Ethiopian state, which explains some of the confusion.
So, to answer your question, had the Roman response to the Kushite raid been to attempt annexation or enforcing a client state with Kush, that would have extended Roman control up the Nile valley, but would not increase access to trade routes to the Indian subcontinent through the Red Sea.
Aksum did not engage in a provocative raid like the example you provided, and Roman trade through the Red Sea seems to have proceeded without much conflict with Aksum. A few centuries after the reign of Augustus, in the middle 4th century AD the Negus Ezana of Aksum converted to Chrisitianity, and in the 6th century Aksum cooperated closely with the Eastern Roman empire in their conflicts with the Sassanid Persian empire.
1 Aksum and Nubia: Warfare, Commerce and Political Fictions in Ancient Northeast Africa pp 140.
There was an existing sea route to India via Myos Hormos on the Red Sea which could be taken overland to the Nile (and up to Alexandria). Ethiopia doesn't offer any great savings since there are six cataracts between Upper Egypt and Sudan that would require cargo be portaged. Furthermore you'd need to caravan the goods from the Red Sea coast to the Blue Nile, increasing your costs. All of that would require garrisoning, increasing your costs even further.
I have a question regarding the sea routes. Didn't the Roman empire control most of northern egypt at the time? I thought they could get the sea routes from the sinai/suez area?