I've found that 10 different professionals will give you 10 different answers and suggestions! Myself included!
I generally only make recommendations based on what I've seen work....and Birkenstock sandals do work to give relief to PF (and a lot of other problems as well). The reason for the pain is the stretching of the plantar fascia and the resulting pull or tear away from the heel bone. I've found the best way to give relief is to support the arch (plantar fascia) which stops the pulling while weight bearing, and thus allows healing to take place.
In running shoes, they generally have a bit of support, that's why your feet feel a little better. Usually Reebox and some Nike styles have a thick foam pad under the arch which again prevents the plantar fascia from pulling.
If you get a chance have a look at our website www.shoetree.com. It describes the condition and how to get relief.
So to answer your question: If you've tried Birkenstocks and they feel good...then wear them! Most people find they work wonderfully!