i've never had a major crash with a car (thankfully), but what i have done in my only crash where there was actual damage, although minor:

(only if you have no injuries) in a friendly way, say that the damage doesn't look bad and you just want your bike repaired so suggest going to a shop and getting an estimate and then handling it not through insurance and just paying the bill directly. in this case i had the right of way and a guy at a stop sign was craning his head looking for fast-moving cars and rolled forward. i was watching and so stopped and dismounted but he ran over my front wheel. anyway wrote down his info told him i would get it chekced at a shop and then i went to a bike shop and picked out a new wheel and then and we met and he gave me like $150 (it was a while ago so maybe it was less i was a poor college student so i was just happy to have a new wheel - i think i added like $10 for my time and hassle which i felt was reasonable (i could have picked out an expensive wheel but i didn't) ) - anyway, probably saved him from having his insurance premiums raised much more than $150 in the next few years. if it turns out to be more substantial damage or the person thinks the bill is too high or whatever then just go through the regular insurance claim process. if this works, a good thing is 1) it's easy and fast as there's no paperwork and you get your bike fixed ASAP 2) if the person is reasonable and doesn't want their premium to go up you get all covered. if it actually goes through insurance they will look for any technicality (my friend had only a rear light and no front during dusk and the insurance only paid 40%)

i agree about always getting the driver info in a crash - swelling and many other injuries often don't show up until a few hours later. i always get the info although i have rarely done anything with it --- but it would suck to find out you have a bone fracture or a major neck/spine problem and not be able to file it through the guilty party's insurance