I couldn't help but notice the striking etymological similarity/connection of the country name 'Osterreich'(with its older form Ostarrichi, that means Eastern Realm) with the theonyms Eostra and Ostara. But the more striking similarity is between its latinised(!) form 'Austria' and the Jacob Grimm's reconstructed proto-germanic 'Austra', also with the Norse dwarf's name 'Austri'(that just simply means "East"), and the proto-nordic word for East - 'Austan'. Also interesting thing is that the Celtic populations called this land 'Norig', no- in this word meaning "east", "eastern", and 'rig' - realm. There even lived a tribe called Noricum.
I have noted that too, but held off on asserting it publicly because, well, scholars will point out that the Ost in Osterreich and Austria simply means East, as in the lands of the Eastern Empire.
However, there are those who contend that Holland is named for goddess Holle. Also, places named for gods were known all across the Germanic lands from Scandinavia over to England; i.e. there is precedent for it.
Also, Ostara as a Spring goddess was likely also a Dawn goddess, so she has an association with the East, as represented in her name. Many tribes associated themselves with gods, and were named for the god they associated with, so the concept of "the Eastern tribes" associating with the goddess of the East and the Dawn is not ridiculous, in my opinion.
The earliest reference to Osterreich is somewhere around the 12th century. So, of course, scholars won't entertain the notion that the name existed prior to that. This is dangerous territory, as I prefer to live grounded in truth, science, and history. But, I'm a conceptual thinker with the depth to understand that just because we can't prove something (yet) doesn't mean it didn't happen or doesn't exist (as the case may be).
So to answer your observation, I'm on the same page as you. But, a solid association hasn't yet been proven, and there are currently no scholars who would agree with us. In sum, I like to think this is true, but I can't say with any level of certainty that it is.
Incidentally, Ostara, Eostre, Ausri are also related to the Baltic Ausrine and Greek Eos, which also relate to "East." :-)
And my article on Eostre