I've never seen a .mobi ranking on desktop (or at least I don't remember having seen one), so I'd assume that .mobi only ranks on mobile devices.
if only the .mobi TLD is available but the other popular TLDs (com / net / org) are taken it means that there's no need for .mobi because - like you said - the exact match of these domains is only good for desktop users which further reinforces my assumption that .mobi only ranks on mobile devices.
So, to answer your title question (based on the conclusions that I've drawn above) I'd say that yes, .mobi does affect ranking on larger devices (if by "larger devices" you mean desktops), and the way in which it affects ranking on larger devices is by not ranking on desktops at all
That's the logical conclusion that I can draw, but if someone knows what happens in reality, by all means, do correct me!