He has always been right wing. People forget that he was picked as a VP for Obama because Obama seemed “too radical” to the DNC. He was at best a centrist, at worst a 1990s centrist, which is to say, a Republican. All you have to do is look at his voting record.
I'm gonna personally beat everyone's ass if we're just caving to the right on this shit. I am not handing it to Bush Jr., Bush Jr. voters, or any of the other fuckers who've come since
"Well, we're gonna lose the left vote because we love genocide, so we should start courting right wingers who are just going to vote republican anyway"
It's extra sad because no matter how transparently they sell out their ostensibly liberal principles the libs will crawl out of the woodwork to scold you for remembering what they used to say and wanting literally ANYTHING from these fucks.
It was less than election cycle ago that Trump saying these exact things and wanting these same policies were reasons to vote for Joe. Because who would allow someone who wants to shut down the border "so we can figure out what the hell is going on" to become president?
They've reverted to a feudal mindset. The democrats have no obligation to their voters. Only the voters are obliged to swear fealty to the Dem candidate regardless of who it is and what they do.
At least that means they will stop using the "wE can PuSh HiM lEfT" line they kept trotting out last tune around, it's too incredible for even them to believe. I very much prefer him running with no platform and just shrugging off questions about why anyone should do this again. It's more honest at least.
"we're gonna elect a conservative and then push him left by defending his every action tooth and nail and promising to vote for him in the future"
Incredible gullibility
I mean, one of his first acts as president was to open more concentration camps. Biden is also a right-wing extremist. He's just a politer, less blatantly unhinged one. The idea that the US is a democracy is such a joke. We criticize states like Russia for having sham elections, but ours are a sham as well since we have no choice but to vote for Democrats. There's no left wing or progressive representation at all.
When I read the title, I immediately expected there to be at least one shitlib coming in here to defend fascist rhetoric just because it was said by Biden. And I was right. 
I don't understand how illegal is a slur...
You don't see why calling people "illegals" is a slur? The fuck?
Because humans aren't legal or illegal? Because it's exclusively applied in a 
way to people who resemble a certain ethnicity? Because it's dehumanizing language that works to subconsciously justify treating them as criminals even if they didn't do anything wrong?
It's language used to criticize an entire group of people out of hand.
Words become slurs as soon as assholes start using it in a derogatory fashion. It doesn't need to make sense, it's just how it is.
Retard used to be a medical term. Then assholes used it as an insult so it became a slur. Then it needed to change to mentally ill. Now mentally ill is started to go out of fashion.
The n word used to just be a term to refer to black people, then assholes used it as an insult. Then assholes started to just use the word "black" as an insult so we needed to switch to African-american.
Language evolves, and assholes ruin things all the time so we have to continuously evolve and change so we don't use the same words as assholes.
I don't think this is the same for something that is:
and
and, in part,
We can keep words to say that people have made a choice without needing to change the language every 3 years. Felons are people that have committed felonies. Criminals are people that have committed crimes. Terrorists are people that have committed terrorism.
Your other examples are slurs because they're used to insult people over something that's out of their control and isn't actually a bad thing. Racial slurs are slurs not because they identify someone as a particular race but because using it indicates the speaker (incorrectly) believes there's something "bad" about that particular race. And, race/sexuality/age/etc... aren't even a choice, so the slur user is insulting the person about something that they can't even control.
Of course, there's a debate to be made that illegal immigration was the "right" choice for that person at the time to escape their situation or that some undocumented immigrants didn't have a choice (children or trafficking victims), but at a basic level we're talking about people who have immigrated illegally, by choice.
so we needed to switch to African-american.
Which leads to ignorant shit like calling people in other countries "African American". That was a dumb choice.
As a rule of thumb, any time you use an adjective about someone as a noun for them it can be dehumanizing.
The baggage with the word illegal is on top of that.
Do we have any words that left that aren't slurs?
Yes, lots of them
"The," "Is," "Of," "At," Most of the articles, actually.
Misspeaking under the duress of the state of the union address is a far cry from "rightward lurch". I'd have to see an ongoing pattern to come to such a conclusion. But you do you.
Duress?
"Psychological pressure" like public speaking for someone with a speech impediment.
He's tried to justify his fascist speak later. It's a freudian slip, not a duress thing.