I think that came down to your first point of them being unsure if they could use Supergirl, and they needed a reason for her not to immediately solve every problem.I kinda get the feeling that they weren't sure if they were going to be able to use Supergirl, so her part was added rather than being essential to the plot. And it was kinda annoying, because there were a lot of moments that could have used her abilities.
But there was one thing that bugged me most, and admittedly, I've never watched Arrow or The Flash, but all of Supergirl and most of the first season of Legends of Tomorrow, but
What the hell was up with Ollie's bullshit, "I'm not feeling sure of myself, so you need to be side-lined, Kara"? What logic was that? You want the most powerful person on your team to step back because you're feeling impotent? I really, really wanted Kara to super-kick him in the nuts right there. And I know Barry wasn't reliable after "Flashpoint", which they mentioned 50 MILLION TIMES OKAY WE GET IT, but he was clearly smarter than Ollie to bring her in. Ollie needs to sit back and shut the hell up, angsty Batman-wannabe.
I liked the crossover in parts, but not in others. I think they can and will do better. Right now my favorite crossover of any of these shows is still the Supergirl episode with the Flash, because putting Kara and Barry together is like putting two puppies in the same room.
I don't get why people hate Felicity Smoak. Having not watched much Arrow, she's mine and my wife's favorite Arrow character. I know people criticized one set of episodes of Arrow being the Felicity Smoak Show--I'll need to find out which season that is.