It is misleading, but no. Warriors and healers aren't women's cup of tea in Tolkien; Eowyn's shift from one to the other doesn't mean she "degraded" herself or such. I like to think of her shift as the transition of the whole atmosphere from defending themselves with the sword and shield to healing the wounds of battle.
Of course there are oppressed women, like Erendis and Rian (who was more pathetic, in my opinion). And I don't deny Eowyn had her oppressed moments too. (Like, the way she got stuck with a sick Theoden, and her powerlessness when Theodred died and her brother wasn't there. But remember that the good guys do have their share of oppressed moments too.) And of course there'll be arguments that perhaps Galadriel isn't really a feminine woman. But for now--I'm writing a paper due in three hours!--I'm gonna have to say,
Eowyn's shift from shield-maiden to healer, her shift from Theoden's caretaker to a lover of Aragorn to slayer of the Witch King to Faramir's wife doesn't make her any less valuable, any less a hero (or heroine, if you will).
I don't think Tolkien had contempt for feminism like, say, August Strindberg. Think Morwen, Ancalime, Luthien (arguably!!). Tolkien may have a dearth of female protagonists, but that doesn't really prove an author's, to use your word, contempt for feminism.
(Someone please post the link to the Ooh La La Luthien thread please. I think it's pretty much related. Sorry I haven't got time!)