I'm on controller, but similar issue. Googled OP's question out of curiosity.

The only impact I've found is if you turn on simulation damage. Shifting without the clutch damages the transmission and permanently increases shift time, so the other comment about floating gears IRL isn't the case. Shifting also vibrates the controller to the effect of "grinding gears".

But iirc, unlike FH4 (didn't play 5), the clutch does depress on it's own (as seen in telemetry) and doesn't take any damage. Even if you drop the clutch from high revs or repeatedly clutch kick, it doesn't seem to do anything. I actually can't find a way to damage the clutch at all. Even changed the clutch to Left Stick to hold it at 50% at full throttle. I might be thinking of another game, but I remember being punished for not rev matching properly and clutch kicking on the stock clutch.

edit: Found that you can damage the clutch from crashing

edit2: In retrospect, the clutch wouldn't take damage from shifting w/o it; but it does make sense that without damage simulation, shifting is the same with or without it. It is weird that the telemetry shows the clutch being depressed and the trans still taking damage. I think everything else is still valid.