I don't think you're being over-cautious.
As well as screws, there are moving or movable components and assemblies in the camera - mirror mechanism, shutter, sensor etc. - that I wouldn't want subjected to extended periods of vibration.
When I'm in the car, I always have my camera and lenses in a padded camera bag, on one of the passenger seats - and usually with the seat-belt through one of the bag straps to stop it flying off the seat if I need to brake hard (both for the equipment's protection and my own
).
That said, I suspect most modern cars with carpeted foot wells and carpet mats wouldn't transmit much vibration to equipment... but why take the risk?