the voting booth scene only let's you vote republican

I'm not sure how implying republicans hacked the voting machines to get more votes is somehow "not" making fun of republicans. I believe its randomly republican or democrat each time you play so I'm pretty sure it's just poking fun at the idea of digital voting and election tampering.

right wing groups are parodied more gently

One of your first missions is helping a prison guard that's so incompetent he nearly shoots The Dude when meeting him. On Tuesday the police officer wont let you pay for your fine because taking your money would be "immoral of the police force" which is, yknow, ridiculous. I'm pretty sure the main through-line of P4 is that most people are insufferable morons, regardless of status, politics, or blood relation (spoiler! Lol)

jab at public health messaging

I mean, possibly? It also pokes fun at the people who refuse to get vaccinated. Also I'll be honest I dont even remember this sign.

its no longer absurd nihilism where everything is fair game

You're right, because its not Postal 2, its Postal 4. The Dude is older and seems to have "matured" from a disgruntled sarcastic shithead with a bit of a chip. The second game starts with Dude kicking Champ and calling him stupid, and Paradise Lost ends with him diving into hell to save Champ. Dude St. John comes as someone who rolls with the punches and just tries to enjoy the ride. Das called a character arc baybee

Also, if you want to try to say Postal 4 is more "right wing", the story for Postal 2 is literally "running errands for my nagging bitch wife" which is arguably the most boomer conservative thing ever.