In addition to everything I already said to defend the Zerk, I'd add a little something : it's perfectly normal to feel weaker as a melee-centric perk when the game is all about guns (or at least "ranged" weapons, to put it broadly). When nearly every other perk can keep a safe distance against the zeds (even shorter-ranged ones like the firebug and support), of course the Zerk will feel a tad weaker... Because you're always gambling your health away, even if you don't care about your killstreak. We already discussed at length that the Zerk does have a few tricks up its sleeve to not just die on the stop as he tries to bash a zed's skull. But it's clearly not a beginner-friendly perk, and you're going to tank some serious damage. The trick is to know WHEN to strike and when to fallback. And yes...Sometimes you might have to fallback earlier than expected.
It's no wonder why the perk got a ranged option from the get-go, in the shape of the VLAD : going full-melee brings you at a disadvantage. Having a little something to rely on when things go south is almost absolutely needed unless you got a very good team, or unless you're being very cautious.
I do believe that the perk is fine as it is, I think you got that now. But I also understand that you might find it pretty hard to play in its current state. It's probably my lest played perk, and I cannot hide that it's precisely for that reason : I tend to misjudge my health pool and go for a play that is just too much for me. But I've also survived ordeals that would have smashed any other perks...
The bottom line of all that is pretty much that the zerk ain't really a PICK class, but a tank one. Making it tankier still? Fine. But then it should have another little nerf somewhere else (being less good at CC, doing less overall damage, having heavier weapons...) to fully commit to that tanky playstyle. Otherwise, it's only real downside would be to have limited range... Which can already be mitigated with the VLAD, Piranha, Frost Fang and Eviscerator.