Yeah, I don’t think it’s the natural result of “putting characters together” - most other creators don’t go quite so far into “Batman must be a privileged dick to stand out among the other heroes” as Timm could shove him, and I’d note that Timm was also doing that in TNBAS more so than he would later, which says something.
Really, “Batman can be a dick and get all the girls” is just the equivalent of “Tony Stark can be a dick who creates half the baddies and still be viewed as a father figure” in the MCU; it’s a formula that appeals to the creators and they just don’t stop running with it. The difference is that DCAU Batman had a contrasting start that was just as if not more successful with the larger audience in BTAS, and that DCAU Batman doesn’t get either Superman or Wonder Woman elevated to a Captain America level of a contrasting paragon who can highlight how he can be part of the problem, like Tony had.