Why do we believe Mars had rivers of water in its past, if its mass is unable to support an atmosphere that would allow water to exist as a liquid?
The conditions on the surface of Mars have not always been what they are now. Liquid water on the surface is not possible in the present because Mars is too cold and its atmosphere is too thin, as you mention. But this hasn't always been the case.
Early in the history of the Solar System, Mars was much more geologically active than it is now. This is because at that time it still had some of the heat it got from the formation of the planets. Mars does not produce heat by itself and its small mass compared to the Earth's is not enough to sustain tectonic and volcanic activity. This is important, because without volcanic activity there is no new carbon dioxide being injected into the atmosphere. Therefore, since the volcanic activity stopped on Mars, the amount of carbon dioxide in the atmosphere has dropped. This has two major consequences: lowering the atmospheric pressure and reducing the green house effect (which was responsible for keeping the surface temperature above freezing). Therefore ever since the volcanic activity has stopped on Mars, the planet has been cooling down and the atmospheric pressure has been decreasing, which made it impossible to have liquid water on its surface.
This is why at some point in the past Mars was warm enough and had enough atmospheric pressure to have liquid water on its surface, but not anymore.