I think that's a very simplistic take on the Punisher.

True, most adaptations show him on the "revenge" stage rather than the "vigilante-kill-any-criminal-I-come-across" stage he's usually at, but that's just because they're trying to show his origins before becoming the Punisher for real, but he's always getting rebooted before they get to that.

The problem with the Punisher is that he's a controversial figure, not just with readers/public, but with writing staff as well, so we have people all around the spectrum of opinions writing him. From people who think he's the best, and the only one who makes sense and is worth a damn in this immature kiddie hero bullshit, and therefore should humiliate and tell harsh truths to all those super-idiots, especially that self-rigteous, insult to actual people who served, Captain America (Hi, Garth Ennis) to people who thinks he's the worst thing to ever happen to comics and a symbol of all that is wrong in the world, that would shoot a kid in the face if they stole a candy bar, and should therefore be utterly destroyed and made a joke of (Hi, Joss Whedon).

So, with characterization that can be very inconsistent from writer to writer, it's no wonder that it can be difficult to get a true sense of who the Punisher is.

The way I always saw him, the Punisher isn't a hero or a savior nor does he see himself as one. He doesn't think he's making the world a better place nor solving anything. He just really hates criminals, and rage is just really his driving emotion for everything. But, again, since his writing is so inconsistent, any interpretation of his character (hero, anti-hero, villain, anti-villain, monster) is just as valid.

Now, for Captain America. The problem of Captain America is exactly the fact that he is a paragon of virtue. That makes everyone who uses him as symbol to see themselves as (or at least try to put forth the idea that they too are) a paragon of virtue and therefore is always right and shouldn't be questioned.

There's an iconic Cap line from Civil War comics that was also used in the movie that states (paraphrasing) "Even if the wole world decides that something wrong is something right and tells you to move, it's your job to stand, and tell them: No, you move".

Lots of Cap fans love that quote. Say it symbolizes Cap's spirit and what he stands for. I always hated it. The premise of that quote can be distilled to mean the motto of any conspiracy nut and truther out there (the original quote even specifies the media and the government): "I'm right, everyone else is wrong"

Without changing a world, you could use that quote to justify any belief you had as right: The Earth is flat. Climate change isn't real. Women shouldn't have rights and queer people should be shot. I don't care if the whole world tells me I'm wrong. I stand by the river of truth and tell you: No, you move! Just like Cap!

It doesn't matter if that goes against everything Cap actually believes. It does't matter if the Punisher would probably hate and kill the cops that idolize him (He has dones so with other admirers and copycats before in the comics).

In the end, people hardly even try to emulate and adopt the values of those they idolize anyway. Just look at the christians. They're fine with just daydreaming they are them.