But in certain places the optimal speed to drive at ... may exceed the legal speed limit

I'll make a stronger claim: in many places (e.g. US highways) the optimal speed to drive does exceed the legal speed limit for the simple reason that the probability of an accident is a function of the difference between your speed and the speed of the traffic around you, and the traffic on interstates generally goes faster than the speed limit.

I think I have seen it claimed that (because driving faster is more dangerous) driving faster, especially near or above the speed limit, is generally a net expected loss in time when you offset low-probability long-time hospital stays against high-probability short-time improvements in travel time.

Back of envelope: your overall accident risk per mile driven is on the order of 10^-6 to 10^-5. Suppose the speed limit is 60mph and you drive for a mile at 70mph, and suppose this gives you an extra 10^-6 chance of an accident. It also saves you 1/7 of a minute. So it's a net loss if the (appropriately weighted) time cost of an accident is more than 10^6/7 minutes, which is about 100 days.

That sounds unlikely on the face of it. (Not least because maybe a substantial fraction of those accidents are little ones in which no one is hurt much.) On the other hand:

1. Something like 1/200 of all road accidents are fatal. Perhaps ones that occur above the speed limit are more likely fatal, but let's leave that aside. If you expect to live another 40 years, then a 1/200 chance of death is about 70 expected days lost. (It's not clear that those expected days quite correspond one-for-one with days spent in hospital after a non-fatal crash.)

2. It also seems pretty plausible that driving sufficiently far above the limit to need a radar detector actually puts you (overall) into the regime where your accident risk per mile is substantially above the overall average. If it's 10^-5 instead of 10^-6, for instance, your expected time lost due to accident only needs to be 10 days to make this a bad idea; that's substantially less than the "death cost" I just estimated.

3. Of course accidents have costs other than the time you stay in hospital. You may have to pay to have your car fixed. You may have to pay higher insurance premiums for a while. You probably attach disvalue to the pain and inconvenience of injury beyond the time spent in hospital. There may be paperwork that wastes a lot of your time. The injury may have lasting effects that reduce your quality of life. You may injure someone else, in which case you may have a legal fight to deal with and maybe some time in prison, which might be even worse than hospital, or a fine. Etc.

I dare say it does sometimes happen that the overall optimal speed is faster than the limit, but I suspect it's less common than most of us like to think.