I totally know where you're coming from because I have the same hangup. But you know what? Birks are great in so many ways. I have more stylish sandal options that I can wear when the occasion necessitates a more fashionable approach. But if I wore them out and about every day, I'd have more foot problems than you can possibly imagine. When you're doing a lot of walking but still want an open-toe shoe, Birks are a very feasible option, even if they don't feel especially stylish. Frankly, I don't think many "comfort shoe" sandal options are much better, looks-wise. I have the Birkenstock Mayari style, which is a nice change from the traditional 3-strap.
I think yours are working well, especially since they are a bit different than the typical 3-strap option. I agree that you'd probably like brown better, but the black is perfectly fine.
I made peace with my Birks by reiterating that my comfort and foot health comes first, and being stylish doesn't have to be Priority No. 1 for EVERY occasion. (Don't hate me, ladies!) I compensate for my Birks by making sure my clothing looks nice and I appear otherwise groomed. I promise nobody would mistake me (or Astrid) for looking sloppy.
Astrid, when you wear shorts and tees with your Birks, maybe try accessorizing a bit? A simple, casual necklace or bracelet might go a long way to making you feel pulled together and fabulous. I like your idea of the head wrap -- sounds like an excellent way to stay cool, too.
I also think a tiny bit of detailing on your tees would work nicely. When I bought my Birks, there was a 20-something sales clerk wearing skinny jeans, a tee with dolman sleeves and the Birkenstock Gizeh sandals. And I still remember how incredible she looked: stylish, yet very natural and down-to-earth. It can be done, especially if your style persona points in that direction.