We could call this the new 'third date rule'. I like it better than the old one.
I echo PPs who say that if someone can't handle dating someone with a serious mental illness, that doesn't necessarily mean that they are shallow, or mean, or a bad person. Living with a serious mental illness is a major, life altering thing that has a profound impact on day to day life. So is being in a serious relationship with someone who has one, and if someone says that they can't handle a relationship under those conditions, then we should take them at their word, just as if they said they didn't want to date someone who was divorced, or had kids, or or any of the myriad of other deal breakers people have.
Further, when you go on a date with someone, you have no obligation, even for politeness, to stick with it in the hopes that you might feel better about X (whatever X is) after you get to know the person for a while. Even if X is something as nebulous as "I don't feel a spark" or "I'm not attracted to them" or "I hate the way they laugh".
What it does mean, though, is that on-line dating can be difficult and very frustrating for anyone who has something that tends to turn people away on first meeting. That can be a mental or physical condition, or kids, or having a stutter or social anxiety, or being overweight, or middle aged. The ones that are immediately obvious on the dating profile will scare away people before they even meet you, others, at the first date. In those cases, I think you either have to develop a thick skin when it comes to immediate rejection, or try to meet people in environments where they can get to know you a bit before starting a romantic relationship, so they can look past what could be a deal breaker on a dating website. On-line dating tends to be very list oriented, and people tend to quickly drop anyone that they have any doubts about at all.