yes that seems tobe glitchy sometimes. it may update yoru time once and display the time, but then if you beat it again the new one won't pop up on your screen.
the time DOES get saved, it just doesn't look like it.
going out and back in works.
I know how it is, though. You want that INSTANT gratification.
The other glitch I noticed is with engine sounds.
In Bumper view, it plays two sounds, a bass engine sound and a treble engine sound. If you do a run and it takes you back to that Session's menu, the next time you do a run from bumper view (w/o exiting game that is) will not play the treble sound.
If you cycle through the views real quick, the sounds are okay again.