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I am asking to see whether "wouldn't rather do something" and "would rather not do something" can be used interchangeably?
In textbook, they use "would rather not do something" see Cambridge grammar
But in real world, people may use "wouldn't rather do something" see Video at 4:08
It's hard to imagine contexts where a speaker might say I wouldn't rather do that (I can only think of an emphatic refutation when the previous speaker has just said I expect you would rather do that).
In OP's linked example the woman asks Are you sure you wouldn't rather have dinner with your friend? The negation of wouldn't is because what she means is something along the lines of Are you sure it's not true [that you would rather do that]?
Take I would like Britain to leave the EU. If you're not a "Brexiteer", that can be negated into I would not like Britain to leave the EU or I would like Britain not to leave the EU (there's no substantial difference in meaning, just a shift of emphasis from not wanting X to wanting [NOT] X).
But with I would rather die the only natural negation is I would rather not die. Ordinarily, you'd only say I would not rather die when disagreeing with someone who implied you might (an unusual situation! :). Perhaps Do you really want to be kept on life support? Are you sure you wouldn't rather die?
