The song is actually a lot deeper than the chorus makes it seem. He actually trying to reason with the Nazi and say, "hey we're both anti-authoritarian, you're just a fucking idiot for thinking Nazism is the best way to go about it."

Punk ain't no religious cult
Punk means thinkin' for yourself
You ain't hardcore, 'cause you spike your hair
When a jock still lives inside your head

This is saying you can't just dress like a punk and be punk. Punk is an ethos not a uniform.

If you've come to fight, get outta here
You ain't no better than the bouncers

Some people see punks and mosh pits and think punks are all violent. But that's not the case, so don't come around if you just want to fight.

We ain't tryin' to be police
When you ape the cops, it ain't anarchy

But with that said, they aren't trying to be the police. They want to dismantle control, not recreate it.

Ten guys jump one, what a man
You fight each other, the police state wins
Stab your backs when you trash our halls
Trash a bank if you've got real balls

This section is basically saying don't take your anger and aggression out on people. It is the system that got you down, not the other people. Take your rage out on institutions of real power, unless you are too much of a pussy.

You still think swastikas look cool
The real Nazis run your schools
They're coaches, businessmen and cops
In a real fourth Reich, you'll be the first to go

As you noted this is pretty straight forward, those you hold up actually hate you. Scary how accurate that still is today.

You'll be the first to go, you'll be the first to go
You'll be the first to go, unless you think

Use your fucking brain, don't be a Nazi.

This is all just my interpretation after listening to this song for 30+ years. I'm sure there are other ways of looking at it.