First, I'd like to address this:
"Go ahead and keep doing it if that's your thing."
This hurt me a bit. I was merely quoting and defending the words in this context. I can't recollect having used them myself either when speaking or writing before, ever. I am pretty reserved when it comes to using these kinds of words, even though I am surrounded by people using them every day.
Same thing with "But I think that's because people are mostly fine with casual sexism, as long as it's not too obvious". I am, for example, bothered at times when I hear male classmates saying "Get back in the kitchen." to female ones. It's meant as a joke, it always is, but that doesn't sit right with me.
I look at the individual, not the sex of a person. I consider "bitch" to be a word free from association to the female sex, today. One thing that may be influencing me is that nearly all of my classmates are male, which means that the times I do hear "bitch" it's used with a male... and certainly not in the way you suggest, except for the last one. Which brings me to the point I was trying to make:
It's an expression for "being a pain in the ass", something which both men and women (and all in-between) can be. Leaving the gender fully out of it is a good thing, as someone else has said, the meaning of words can change.
Finally, perhaps there are a few cultural differences. I grew up and live in Sweden, and while we borrow many words from English perhaps not all their meaning, and in particular, the history behind their usage is brought along with them. The old fashioned use of bitch may be more alive in other places, I suppose.
In short, I do not appreciate words and expressions where the downgrading of women is obvious, but I do not consider bitch being one of these words. If you do find someone using it to downgrade women, teach him (or her) a lesson. When I read "And yet, why don't we hear the people who always bitch about Google complaining about Microsoft?", I didn't get an image of "women in their right place" in my head, but one of raging nerds.