Getting mouthy with teachers and struggling socially to me are two totally different things.
I have six year old daughter who is extremely polite and respectful and would NEVER do this, BUT, and this is a big BUT, we disciplined it VERY firmly a few times in the past when she attempted normal disrespectful speech to a parent or a teacher or talking in a "bratty way" to other kids. It happened a couple of times when she was new in daycare (3-ish)-natural of course to be around lots of other kids smarting off and snapping at each other. We explained what it meant to speak disrespectfully, spelled out our expectation and enforced discipline. Then it happened in normal scenarios at home or in gymnastics class and a couple of "short" remarks to her piano teacher. Each time I reminded her what was expected and enforced discipline. The last time was when she was four and was getting rewards taken away in K4 for pushing boundaries and talking back.
I came down like a ton of bricks because at four she was well old enough to be disciplined at home after the fact, so I told her I would speak to the teacher EVERY DAY and if I EVER heard she was being disrespectful to the teacher she would be spanked immediately when she got home and lose her favorite item for a whole week (she has a blanket she literally cannot live without at night). She knew from past consistency it was true and it never happened again. That was almost two years ago. She also never talks back to us. I also always speak to her respectfully, we have an extremely close and affectionate relationship, I never yell at her, and she gets genuine praise all the time for her achievements and for just being her because I love her, but if she is ever rude or disrespectful-her world stops. It's THAT black and white. And that's why she doesn't do it.
Now believe me, she's allowed to have her emotions, to be upset, to disagree, and to stand her ground on things-we just draw the line at disrespect to teachers and others in the specific scenarios where that is what it is.
I'm skeptical of the psychologist angle only because SO MANY kids I see in all my kid's classes act exactly like you daughter is acting. My kid's Tae Kwon Do Teacher has like five kids in his class of 10 who talk back to the teachers and snap at each other. My daughter and son would be doing it too if we had not been VERY TOUGH on it. My 2 1/2 year old is now learning with discipline not to say "no" and talk back to parents the same way.
We do all the same things you do, family time, healthy activities, family meals, etc but ALL kids push the rude and disrespectful boundaries. Some more than others. It takes discipline to enforce right behavior unless the child is born naturally angelic. If you both work full time, she may be crying out for more attention, and you should provide that at times aside from discipline scenarios, but don't let it blur the line of enforcing good behavior firmly enough.
So definitely make sure nothing is more wrong with the psychologist, but don't let them treat your daughter like she has a medical "problem" and make it worse by giving too much attention and not recommending enough discipline. Use your gut. As for struggling socially, I wouldn't worry too much if she is able to play with other kids etc. You'll know if she's showing signs of something "wrong". Just being mouthy and disrespectful is pretty normal at this age. Again, lots of my daughters classmates act this way. They're like little kindergarten mean girls. Good luck, and good work acknowledging this and working on it! This book is great for nurturing a grateful attitude, good spirit and respect in kids this age, Back to Basics Discipline by Janet Campbell Matson.