With berserkergang, they heal 40 HP per minute. That means they heal 2 HP every 3 seconds. An elite skirmisher has a reload time of 3.0 seconds, as well as the time it take for the projectile to actually hit. With all armor upgrades for the Elite Berserk, it has 5 pierce armor. The elite skirmisher deals three base pierce damage, but with all three attack upgrades and chemistry can get this up to 7 base attack. Maths then means that it will do two damage to the Elite Berserk every 3 seconds, which gets exactly countered by the healing. If you add in thumb ring, then it fires faster, but the berserk will still normally kill it before it matters. The projectiles are also rather slow, which further decreases the rate at which the Berserk takes damage. Also, if it misses any of the attack upgrades or chemistry, then it only deals one damage every few seconds, meaning that Berserks can handle it, even without Berserkergang, because then they heal 1 HP every 3 seconds, although you do again encounter the issue with Thumb Ring.
Civs missing any of these techs:
Celts, Cumans, Malians, Persians, Slavs, and Teutons all miss Bracer.
Aztecs, Bohemians, Britons, Burgundians, Burmese, Celts, Franks, Goths, Khmer, Sicilians, Slavs, Teutons, and now Vikings all miss Thumb Ring.
That means that out of the 39 civs, 17 of them will never be able to use Elite Skirmishers to kill Elite Berserks with Berserkergang, and 3 of them will never be able to use Elite Skirms to kill any Berserks, with or without Berserkgang.
Note: All my maths assumes that the Berserks have all the armor tech researched.