It's been a while since I had a baby, but aren't car seats supposed to be installed
without the baby in them, so you can put your knee in the seat and push it down so it's taut? If so, then I will gently point out that you and your husband aren't perfect either, apparently.
And have you considered having her, say, bring the baby in the house first, making sure she's safe, then going back out to get the stroller, so the baby is not actually
in the stroller when it's being brought up and down the stairs? Honestly it sounds like anyone could have an accident in that situation while the baby is in the stroller. There are lots of different ways to look at any given situation.
You mentioned a language barrier. I take it this nanny is from another country? It seems to me there would be a wealth your child could learn about another culture which might "offset" any awkward things she has to say "politically". You will never be able to control everything your child sees and hears. I always felt like things like what your nanny said became talking points for me and my children. They developed a well-rounded view and were able to think through both sides of an issue from hearing all kinds of crazy things from a good variety of people in their lives. Ymmv
My mother helped watch my kids for me, and I also hired a bunch of caregivers along the way - I had twins so I always had two people there since my mother was older at the time. I never found a perfect hired caregiver - they all had their pros and cons. Things sometimes drove me crazy, but I knew my alternative was to quit my job and stay home and care for them myself. I tried to focus on whether they were happy and healthy in their care, and that was the deciding factor. I did have to let a couple go, and it was hard, but you do what you have to do. As for my mother being older, she was pretty mobile and such, but she always told a lot of stories from her youth (which could be dicey at times

), sang songs, made them hot cinnamon buns and their favorite foods, and served them with love. The tables are turned now and they are helping care for her. My point is, don't be afraid to let your child be exposed to people who aren't necessarily exactly like yourself. It can be a great experience for them.
Good luck with your decisions. Being a parent is hard, and not for the faint of heart!