It could be a cool plot point for her to have to overcome in time. You can start a character as obnoxious and arrogant and have them grow from their experiences to realize this crusade isn't as easy as it first seemed and come to respect just how much your predecessor sacrificed to wage that war without losing the character in the predecessor's shadow.
Heck, Robin's entire journey to Nightwing is essentially about that.
But I watched Supergirl (admittedly on a 2nd monitor while MMO'ing - how I've watched all of CW shows for years now) until this current season. It's just gotten more heavy handed and hamfisted in its methods and dialogue, so I have a strong feeling Batwoman will be more of that and no character growth beyond what you see in the trailer.
A huge point to me would be that "I'm not going to let a man take credit for a woman's work" when the shadow of the Bat would work to her favor. It worked in Dick's favor all the time. You let them think Batman's back. The trailer shows her in a more Batman like costume until she changes the mask and adds the wig. Criminals would often have this relief of "oh, I thought it was Batman but it's just you" with Dick and that underestimation was an advantage for him. The fact that she's arrogant and wants everyone to know she's not Batman could actually bring about storylines that work against her and she has to look at things differently.
And the show is going to have a hard time escaping that shadow, regardless.
Where is Batman? You have to have a damn good explanation for why Batman, and Bruce, abandoned Gotham for 3 years now.
Where is Dick? Bludhaven was mentioned way back in the second season of Arrow and Batwoman (and thus Batman) exist on Earth-1, so it's not a leap to expect Nightwing is out there, but for Batman to disappear and Dick ignore everything for years seems odd. Having Grayson FINALLY appear on Batwoman and highlighting the differences between Bruce's training Dick as a detective whereas Kate has more focus on military & survival training would be a lot of fun.
We've heard Harley Quinn's voice on Arrow, so we know she, and thus logically Joker, exist. Would Joker take kindly to someone suiting up as his nemesis? We saw that this Earth has a twist on Mr. Freeze now likely being Mrs. Freeze as the freeze gun was labeled as Nora's property in Arkham. The trailer is kicking off with this Alice and the Wonderland gang....surely Jervis Tetch will be brought up, right? But that draws again on Batman's villains, keeping the show somewhat overshadowed by the lack of his presence.
The ongoing "Batman without Batman" is going to get frustrating eventually. Gotham, Titans, Pennyworth, Batwoman.
Bite the bullet and do the damn thing on AMC, Amazon, HBO, or somewhere that will put the budget in for a big spectacular Batman series with season long mysteries running across the episodic run ins with the rogues gallery with a growing Bat Family over time.