I live in the OWE and my aunt lives in Birkhead. The roads are a fucking nightmare. And I agree, the return on investment is not good.
The plus to the OWE is it's size. Want to go for a jog or walk your dog? You've got many acres to safely traverse. In Birkhead, unless you want to walk the much smaller neighborhood, you're subjected to the insanity that is cherry street.
While Birkhead is super close to the OWE, I'm always shocked at how "hood" that place feels vs. the OWE.
Birkhead has the benefit of the fence, but properties on the fence line often get their back yards / garages etc messed with. Even being so close to Central catholic doesn't seem to help as much as you'd expect.
Also, you're much closer to the hospital so you can add on the added traffic noise / sirens etc.
No question I'm biased, but if I was going to live in the area, i'd pick the OWE over birkhead any day of the week.