If I understand it correctly, when messaging girls in open relationships, the usual dynamic is reversed: rather than having to convince them you're looking for a relationship and not just for sex (which I'm not going to do because it isn't true), you have to project that you really are looking for something casual, and not looking to impinge on their relationship.

That's not true of me or most non-monogamous women I know, although I'm more familiar with people who identify as "poly" rather than "open". For me, if I'm dating, I'm usually open to both a romantic relationship and a friends-with-benefits sort of situation, depending on how we mesh when we meet in person. I'm not into one night stands and I prefer to be approached on friendly, non-sexual terms. That doesn't change based on whether I have someone else I'm serious with. What I do look for when I'm dating someone else is question answers that demonstrate comfort and familiarity with non-exclusivity.

Some people for sure have a "you can have sex outside the relationship but don't fall in love" policy, but I don't think it's as common as you're making it out to be.