Does EASA have a problem with drunk pilots? Especially private pilots, who are impacted nearly the same by the FAA's rules. Because an EASA medical is a medical, not criminal, check. The FAA extrapolates an incident of DUI as a diagnosis for alcoholism, and then maintains that diagnosis until you complete a program of recovery that is so specific that there are only a few dozen of some of the specialists they send you to. Their preferred program, AA, has little to no scientific evidence beyond the benefits of a support group in general, but if you didn't follow the steps then according to FAA you aren't 'really' sober.
So, adopt EASA rules. Or if you're not going to trust me when I say I don't drink, fine give me the random tests for 6 months to a year. Or accept treatments like the Sinclair Method with are proven effective 80% of the time, 10% of those failing simply aren't following the treatment. HIMS claims a failure rate of 7% but that's with the banhammer of the FAA standing over participants, so hard to say how effective it really is otherwise.