While I had several problems with the A or B dialog options in the DLC, I have the feeling you are missing some of the background lore of the game.
Night City is an independent state - it literally has border control between it's region and the NUSA. Myers has no authority in NC, if I got that correctly. Sure, NC government/politicians would help her, at least to dodge a diplomatic sh*tstorm.
She, and later Reed actually need to rely on V, and they know that. V has connections in NC, free movement in and out of Dogtown. Whatever they think about him/her, they have to avoid alienating him - this especially goes for finding Songbird, and V's connection for her. Behaving like a**holes with V would've made them look like idiots. It would have contradicted their interests. (I'm talking about behavior, not what they think!)
With that being said, Myers's character was just a cliché corrupt politician, who will act righteous while they can, but will use any method to save their a**es. My problem was how the GAMEPLAY was forcing her to be seen (by the player) as a girlboss. She drops V on his/her a**. After the explosion at the plane, she gets up first - even though she was closer to the blast. V can throw around people and rip doors from their places, but will need her help at almost every strength requiring action!
Also, she does all this and the combats, running and rolling IN F*CKING HIGHHEELS! (This was more likely just a design oversight, still annoying.)
What we got at the oath is the option to create V's stance - for either option - on the unification/NUSA government topic. It's also game lore that unification - as the Unification War - is a topic that has both supporters and opponents in Night City. V's opinion on this topic - as far as I can remember - wasn't mentioned in the Base game. You basically decide what he/she thinks about those things there.
I agree that this topic could have been explored far better (and for me, would have been more interesting than what we got), but the oath part wasn't as screwed up as other parts - in my opinion, of course!
V's behavior in the DLC is problematic, on that I agree with you. While this govenment/spy is totally new to them, they could have had more options to remain skeptical! His/her attitude towards Songbird up to that point you mentioned felt soo forced for me, I couldn't wait for her to go away.
This is my take on what you wrote (hopefully I managed to point out relevant things), but, by all means, feel free to disagree