The flexibility of the car classes is designed so that you can race a wide variety of cars in whatever class you enjoy racing in. However, you shouldn't necessarily upgrade every car all the way up. Just because you can upgrade some "D" class car doesn't mean that it will be competitive as an "A" class. To really determine what class you want your car at, you have to examine the other cars you will be racing against and try to beat their stats not just their class. Also, it's very interesting (to me, at least) to race in lower tiers because the cars tend to be very closely matched so precise driving becomes important.
Just to note, the reason that cars don't perform well in every class is because each part contributes to the overall class for the car, but they don't necessarily contribute the same boost to performance.
All that said, if your difficulty is not set that high and you want to go fast, then go crazy with your upgrades and you should be competitive for the most part. If you're racing at the level where you must match yourself against the other cars then tuning set ups will be more important than the class that you upgrade to.