Is it ok to drive car immediately after starting the engine?

I believe the owner’s manual instruction is intended for people who would normally give their cars a lengthy warm-up period, perhaps a minute or more. On modern cars, a warm-up is not necessary.

The 10-second suggestion is just that – a suggestion. That’s about how long it takes to fasten the seat belt and to adjust sound and air. Driving off in fewer than 10 seconds is of no consequence.