I'd be concerned that would put too much emphasis on the fastest lap, turning races into hotlapping competitions instead of focusing on the main objective of gaining/holding position. Depends on the combo I suppose.
Although I do enjoy going for the fastest lap if I've spun out or crashed. It's nice to redeem yourself after a mistake with having the fastest lap at the end.
LFS tells you who's just put in the fastest lap anyway.
No i don't think so. It's not more like in real racing. Drivers there in a race will also first concentrate on the race and not try to get FL.
It would be just an information who is pushing hard(fastest at the moment) or something like that.
And that LFS tells you is not enough, it is only for a short moment and you can not read in every situation. And in my age even not remember after a few laps
Maybe it could be done with a button!? You press "l" for eg, and then the FL will be displayed again for a few seconds....