It can matter, but it's really not as simple as you propose.
Research shows that for most women, birth control pills actually create a more stable mood. It's part of why women susceptible to PMDD are often put on them. Which is another thing to bring up - it's not like women can't have hormone-related emotional problems without birth control, anyway. Like, dude - even if not to the extent of PMDD, PMS is a thing!
I mean, it's important to realize that the whole way BC works, though, is by mimicking the progesterone that's released in some parts of the menstrual cycle and then throughout pregnancy, and that this progesterone suppresses other hormones that would otherwise orchestrate things such as ovulation and the thickening and preparation of the uterine lining by fluctuating. That's the whole reason it works as a contraceptive; Usually there's this big surge of hormones and that stimulates an egg to be released, and then a drop off that stimulates menstruation, and these two points of drastic hormonal change can trigger some emotional changes in women as well. Continuous progesterone smooths that out, keeping everything at a more level state. (And without that surge inducing ovulation, there's no egg to get fertilized, thus, viola, protection from pregnancy).
But synthetic progesterones used in birth controls are not bio-identical to the natural stuff (not that you'd even necessarily want it to be), and they can actually vary quiet a bit in influence over other hormones, acting more or less on different receptors, ect. Some are much more androgenic or estrogenic than others, for example, and depending on the women themselves, this can matter a lot. Also, many common birth control pills are "combination pills" - including a sythetic estrogen in addition to the progestin. This isn't necessary for it to work as a BC, but it was added because it helps keep the estrogen levels from dipping too low in most women (as can happen on progestins) and the side effects that can come with that (which can include low mood! Kinda like what happens in menopause). However, while most women are better off with it, some women are particularly sensitive to estrogen. It's counter-indicated for women with a history of migraines, for one. It can also cause water retention (bloating) in other women (if they are susceptible to such).
There's more than a dozen types of progestins used in BC - we're up to 4 generations now, and they come in different doses and different schedules and different delivery systems (transdermal, pill, implants, shots, ect.) and some with (synthetic) estrogen and some without and, well... honestly? If one kind worked well for all women in all situations, I don't think this would be the case. We haven't really perfected birth control, to be honest. Not that it doesn't have value and doesn't help many women. Personally, I'm waaaayyy better with it because of how it helps my endometriosis. I also save a lot of money on tampons. Win-win. But I'm on the right kind for me and I don't have side effects.
Decision making is an interesting thing to bring up, as there are some proposals that women do seek out a different type of mate dependent on where she is in her menstrual cycle (or if she's in it at all, as in like, if she's pregnant). IIRC, women who are pregnant, or have the hormone profile closer to that state, tend to look for more of a "provider" and "familiar" mate who will stick around and care for their (potential) kid. Women who are on the opposite side of the spectrum (like, ovulating) supposedly tend to seek out more, errr... how do I put this... virile mates? With less emphasis on long-term care taking ability?