What is the term for an argumentative style that relies on immediately extrapolating or threatening worst outcomes? [closed]

You could describe this style of argument as alarmist (adj.). The person who adheres to this style of argument could also be called alarmist as an adjective or even referred to as an alarmist, using the noun sense of the word.

The OED gives this definition:

Characterized by a tendency to exaggerate potential dangers or an eagerness to express fears or concerns publicly; esp. that creates needless worry or panic in this way.

In use in the wild, the word is often somewhat derogatory, meant to describe a person or their argument as extreme, overblown, or highly reactive, which seems to fit with the sentiment sought in the question.

Here is an example of "alarmist" employed with a negative connotation in an opinion piece in the National Review:

Though rhetorically powerful, this alarmist tone is misleading.

• Andrew L. Shea, The National Review. "Private Universities Protest the ‘Endowment Tax’" - March 20, 2018

If you are suggesting that it’s a character flaw, psychological defect or mental health issue, then catastrophising (Merriam-Webster) is the word you are looking for.

Let us start by considering why some people catastrophise – that is, on hearing uncertain news, they imagine the worst possible outcome.
— The Guardian

Another is to title the person (a) Chicken Little, which has roughly the same meaning.

However, the style of argument is perhaps a subtly different thing, and I’m not convinced any one word will carry all the strength of meaning you want. The first two examples are probably best handled with fly(ing) off the handle; the third is more in the category of things I’d assume were intended to be (at least partly) humorous, but you could fall back on my first suggestion if they’re serious.

I would call this "invoking the nuclear option".

nuclear option

The most drastic or extreme response possible to a particular situation.

https://en.oxforddictionaries.com/definition/us/nuclear_option

Here's an example that actually happened to me this week.

I rent a townhouse in a city I visit often. On Wednesday, I received a text message from the owner saying that she wanted to sell the property and wouldn't be renewing the lease. I couldn't respond to her that day. I planned to respond the next day, but before I could I received a second text message threatening me with legal action if I didn't cooperate with the brokers.

She went nuclear in less than 24 hours. It was an extreme and unnecessary escalation. She had never had a problem with me before.

A less formal term might be appeal to ridicule (click for that Wikipedia page). Quote from that page:

"Appeal to ridicule is often found in the form of comparing a nuanced circumstance or argument to a laughably commonplace occurrence or to some other irrelevancy on the basis of comedic timing, wordplay, or making an opponent and their argument the object of a joke." 1

Attribution

1 Wikipedia. (2018, March 21). Appeal to ridicule. Retrieved March 23, 2018, from https://en.wikipedia.org/wiki/Appeal_to_ridicule

When I read the example sentences, they all strike me as making a mountain out of a molehill. From wikipedia:

Making a mountain out of a molehill is an idiom referring to over-reactive, histrionic behaviour where a person makes too much of a minor issue. It seems to have come into existence in the 16th century.

However, based on the title of the question, I'm not sure this phrase would be considered a "term." And while it's exaggerating the situation, I don't think this idiom can only be used for the "worst" outcome. On the other hand, the example sentences don't meet these criteria either, so I'm not sure the "term" and "worst" criteria in the title are essential to the answer.