As far as I can tell, Stilicho was born in the Roman Empire. Was he considered too barbarian? Were there some other considerations? It feels like Rome had a long history of military people throughout the Empire becoming Emperor, but this period seemed to shift more towards having powers behind the throne instead. I'd love to hear some thoughts on why all this is.
He didn't really need to - as Honorius' guardian he held a lot of sway as it was. While his bloodline was not pure Roman, Theodosius I had respect for him as a military leader that he was not hesitant (according to sources like Claudian) to appoint Stilicho as Honorius' guardian.
For Stilicho to have been proclaimed Augustus he would have needed the support of the military, but as far as I'm aware, they were still fairly loyal to Theodosius after his death.
This was a period of time where, with the empire split, there needed to be parties on both sides involved in the politics of governing, rather than just one governing body as had been the case before the empire was East and West. Having Stilicho and Eutropius as guardians gave both of them the power of influence without necessarily facing responsibility.
I hope this kind of makes sense - I did a postgrad presentation on Stilicho last year so I can provide ample sources if you need. A great one to start with is: Stilicho: The Vandal Who Saved Rome. Ian Hughes. 2010. Pen & Sword.
Let me know if you would like further sources!