I have a few thoughts, here. I think, on principle, it is good that you were honest about the anxiety, but in this case the cost was too high (through no fault of your own!) And, unfortunately, the fact that some people may respond this way to your condition is something you have to factor in when weighing whether to disclose or not. I think, given what happened, I would wait to disclose your anxiety until after trust has been built, because it is totally unfair and ridiculous that she called care.com and your account was closed. If she felt uncomfortable with it, then, yes, it is her kid, she has the right to turn you down. But anything beyond that is wrong, in my opinion. I think you could have emphasized to her that you have a long track of excellent nannying and good reviews despite your anxiety. Given how she reacted, I don't think it would have made a difference, but in the future, I would emphasize that.
The other thing is your gut reaction to the father. I think you should trust that, and maybe shouldn't have taken the job anyway. I also think there is a way for you not to immediately disclose your anxiety without also misleading the parents. You don't owe them that information immediately, especially if you were unsure about whether you even wanted the job. I think it would have been perfectly ethical to just finish the interview, go home and think about whether the anxiety that the father causes and the fact that you would be expected to go out in public a lot would affect your ability to perform your job. If you concluded that yes, it would, you could turn down the job and if they ask why, just say 'It was not a good fit for me' no obligation to explain further.
You are clearly a good nanny, and it is a shame this interaction went south. It also sucks that you loved the kid, but the parents were the problem (and I really believe they were super problematic here). How have you dealt with your anxiety with past families? Have they typically known about your anxiety? If so, consider that they can be resources not only as reference on your online profiles, but as clients who can speak directly to how you were able to excel without your anxiety becoming an issue. They can both talk directly about it to future clients who express concern about it, and also just tell you what their experience has been regarding you anxiety, so you can pass that on to the potential client. That way, if someone brings it up, you can say, 'I appreciate your concern, and in fact I have worked with many families without any trouble' and then give specific examples for what people have said about their positive experience with you, and then offer to connect them with said clients.
You have a lot of strengths, so I would play on that, and if you have the feeling a certain family just won't be able to handle or understand that...let them go. Don't give them the opportunity to destroy your reputation like this lady did. :/ You have the skills, so you want to work with a family were there is mutual respect. I think you owe that to yourself.
Also, have you considered contacting your previous clients to ask if any of their friends need a nanny? That was you will already come recommended by a trusted friend.
Good luck, and sorry people suck sometimes. That is on them, not you.