In the Wikipedia article on Anthemius, the following appears:
As Aspar could not sit on the throne because of his barbaric origin, he opposed Anthemius whose prestige would have made him independent and chose a low-ranking military officer, Leo; in the West, as his barbaric origin barred Ricimer from the throne, it was Majorian who received the purple.
That part of the Wikipedia article was attributed (incorrectly it seems) to Prof. Ralph W. Mathisen. I emailed him and here is his emailed response, with his permission:
Nobody's "barbaric origin" "barred" them from the throne. Theodosius II was the grandson of the barbarian general Bauto, and there are several other examples. The Illyrians are not good examples, as they were all perfectly good Roman citizens. Of course, barbarian generals also were Roman citizens. Ricimer and Aspar would have been every bit as good Romans as Aurelian, perhaps even better! On Aspar note a letter of the Ostrogothic king Theoderic to the Synods of Rome of 501 CE recalls, "at one time it was recommended to Aspar by the Senate that he himself become Emperor. He is reported to have given the response, 'I fear that through me a precedent in government be established.'"
As with anything from that time period, it is difficult to say anything for certain. If we can believe Theodoric's letter, we have to assume that if Aspar did indeed fear a precedent would be set, that a counter-precedent of some kind must have existed. It didn't absolutely bar anyone from the throne, but it was enough to make Aspar and Ricimer think twice.
Pure speculation, of course, which may not be appropriate for a Q&A site like this one, but in this case, what else do we have? There was some reason Aspar and Ricimer didn't take the thrones when they could have.