I don't know whether exercise tests have been mandated for class 1 medicals - not an AME. I gather they're falling out of favour in general medicine now that tests such as angiography and perfusion scanning are considered safer and more informative.
I have no expert knowledge regarding any of these issues. But a quick 'Google Scholar' will show up a number of links suggesting they're significant issues. I haven't digested the papers in sufficient detail to fully appraise them, and don't have the time to do so. On the other hand, it's not so hard to get some kind of an overview of the context in which aviation medicals are justified.
Going back to the headings I used:
1) Suicides - a difficult thing to assess for multiple reasons such as a frequent reluctance amongst coroners to recognise suicide where there is any doubt and the ultimate unknowability of what was going through somebody's mind prior to an accident. If they haven't written a note, it's hard to know for certain.
Analysis of NTSB Aircraft-Assisted Pilot Suicides: 1982?2014 - Politano - 2015 - Suicide and Life-Threatening Behavior - Wiley Online Library identifies 51 pilot suicides 1982 to 2014 but there are almost certainly likely to be a lot more.
https://www.ncbi.nlm.nih.gov/pubmed/25199127
Estimated the rate of pilot suicide at 0.33%. Again, I'd wager that the true rate will be considerably higher.
2) Sudden incapacitation
https://crashstats.nhtsa.dot.gov/Api...ication/811219
states that 84% of people in passenger vehicle crashes who were suddenly incapacitated (e.g. faints, hypoglycemia) had suffered previous episodes. That would imply that similar accidents are by and large preventable.
4) I could have worded this a lot better. I don't mean to call people who make mistakes stupid (I've survived a fair amount of my own stupidity, as I suspect we all have), but a lot of aviation accidents are the sort of lapses that in retrospect are going to seem dumb. I don't mean helicopter-towing-speedboat accidents, but regular everyday things like EFATO turnbacks or landing gear-up or proceeding into bad weather. They're going to be classed as 'pilot error' on any accident report but could there be a medical contribution?
The US emphasis on sleep apnoea (and its potential to be an issue with anyone with a high enough BMI) is the sort of thing I'm getting at. If you have significant sleep apnoea, your cognition and ability to make sensible decisions suffer, but because you may always be impaired, your insight into it is likely to be limited.
The full blood count you get as part of a medical will give information on anaemia (may cause reduced performance at moderate altitudes), alcoholism, vitamin deficiency (B12, Folate are important for cognition and deficiencies change the size of your blood cells). Your BMI is easy enough to measure.
It may be that these issues contribute to more accidents than the obvious all-or-nothing medical incapacitation events.
I've heard figures such as 5% of private pilot accidents being due to medical factors - obviously these are 'failures' of aviation medicals. The question is, how many 'successes' are there where people have not died because their AME either stopped them from flying or resulted in them being treated such that it no longer posed a risk to them (e.g. swapping Piriton for a non-sedating drug).
As I mentioned before, I'm on the fence. I think there's a strong argument for solo pilots to do pretty much what they want - perhaps anyone who flies in airspace should have a hearing test and you might have some caveats for some personality disorders and flying over cities. If an ex-WWII pilot with a dodgy heart wants to take a risk, I'm all for that. But for grandparents who want to give joy-rides to their grandkids I'm not certain. I can see that AMEs and aviation medicine really do have something to offer, even for PPLs.
It would be really interesting to see an AME argue the case for their continued existence, head to head with someone arguing for deregulation. Perhaps the editors of Pilot or Flyer could take note?