You can do a more thorough exam, but to what end?

Two obvious concerns, cog/coordination decline and sudden incapacitation.

I tend to think that recurrent sim should catch the former, although if you really wanted to you could use a PC based video game to test basic attitude instrument flying with some other multi-tasking activities thrown in. Of course somebody would make a very close facsimile of the FAA software and everybody would practice before their medical. Such a test could be dialed in to be fairly easy for any normal pilot, and tuned only to flag somebody with real clinical decline. In theory. In practice it might disqualify 5-10% of the pilot workforce, and make another 30% really sweat their medical. Devil is in the details.

For sudden incap, it's a bit easier. Determine the likely causes of sudden incap (should already be well documented), perhaps tweak that to be aviation specific if you have aviation specific data... that would be useful because obviously the current system isn't good at catching it. Most of the news report about pilot incap seem to involve cardiac issues, frequently between age 40-60. I'd guess a lot of sudden incap in the general population is due to diabetics losing control of their blood sugar... we do get tested for that, unless your AME is committing outright fraud.

Then determine what additional tests can practically screen for whatever your top statistical incap risks are. If I had to guess, I'd say a treadmill stress test and maybe cholesterol test. A stress test would actually be good IMO because it would tend to force some folks who really need to into regular exercise. That's good for cog health in addition to cardio, blood sugar, etc, etc.

121 aviation generally is certified with a 10^-9 safety factor... they should probably start with that and work backwards from there. That's how they certify airplanes and operating procedures.