I don't think Doula/CBE experience would be frowned upon at all, but I also don't think it would get you "in" like actually working on the unit in some capacity (OB tech, CNA, unit clerk, etc.).
It can be an adjustment to go from a circle of peers who are educated, healthy, and on the same page as you are about birth- then hit a unit where the vast majority of the patients may be largely uneducated about pregnancy and birth. It can be dismaying.
I pretty much knew it would be that way before I started - and you know what? You really can turn that dismay into an opportunity. It may sound cheesy, but you have so much more opportunity to teach and empower those patients who are not informed! You probably won't turn their entire world around in the time that you are with them, but at least you are not preaching to the choir - you have more chance to make a difference with these kinds of patients than with those who have already read the book & signed the birth plan to the last "T". They are usually very appreciative, too, that you have taken the time to explain things to them (like maybe no one has before!).
It's all shades of grey out there... usually people are doing the best they can with what they know. You mentioned that you plan on doing a healthcare mission, if that will be in a different culture then you'll see even vaster differences in health beliefs... the L&D thing is just a milder version of the same thing, in your backyard.
Does that make sense?