Do you honestly think that Apple next week is going to leave a bunch of people on the public beta stranded on a beta release until the end of September or October? Public beta testers have always been able to update to the new OS and then choose to drop out of the beta program, if 13.1 isn't the public release then that's going to be a lot of people annoyed at Apple.Based on what proof or evidence? This is nothing that suggests this. iOS 13.1 is only at beta 2, it will need at least one or two more betas.
They don't say not to use public beta on a main device like they do with the developer betas.