Word/phrase that represents "unknowingly Arguing against something/explaining something based on false statement X given"

I'm looking for a word or phrase that encapsulates the action of defending statement X of opponent, despite X having no validity, or unknowingly arguing/explaining based on a false statement X. For example:

*Adam, a cat person, says that all dog people must manually lick dogs mouths clean, and this is why dogs are a disgusting pet.
*Jill, who hates cats, but doesn't own dogs, argues against Adam, saying "at least the dogs mouths are clean for it, and dogs allowing people to get that close to them just shows how much they love them. Cats lick all over themselves and are disgusting, no one would lick their mouths. Plus they are selfish".

Notice here, Adam makes a statement that is loaded/begs the question, but Jill doesn't know enough or simply assumes what Adam says is true, and tries to come up with reasons on the spot why X is justified, despite never seeing/experiencing it herself, in the pursuit of showing that cats are worse than dogs.

I'd expect a phrase that follows this to be something like "Falling for the X" and used like so in the following context:

Ruth, a dog lover, overhears the conversation between Adam and Jill, and turns to say: "Adam, No dog owner is forced to lick the mouth clean of their dogs. Jill, I appreciate the sentiment, but you just [Fell for the X]."

In real life I've seen this play out across both science questions and political discussions.

In the recent US presidential election, 538 claimed 20,000~ votes all for Biden in Philadelphia county in Pennsylvania. One person said "Look at what happened in Pennsylvania! Not one non Biden vote? Clearly this shows voter fraud". Other people said "Well people are more likely to vote for biden in mail, so definitely not suspicious!".

In this case, it was basically statistically impossible for this to happen. It also turned out the votes were legitimate, they were just batched by candidate accidentally in the unofficial tally, the next batch had 2x times Trumps normal margins. The people responding assumed an actual batch of only Biden votes really happened, but in order to support there was no voter fraud, they went on to support the faulty assumption that a group of 20,000+ mail in votes was entirely Biden. People were trying to defend the claim that 20,000 votes came in from a normal mail in sampling distribution entirely for Biden, despite the claim those ballot counts were sampled from normal mail in order was completely false.

In a similar manner, this question on physics stack exchange https://physics.stackexchange.com/questions/593122/why-are-red-and-blue-light-refracted-differently-if-they-travel-at-the-same-spee has many answers from people who take Op's assumption (two wavelengths of light will always travel at the same speed no matter the medium), and try to explain why refraction is different only with the equation tautology, implicitly affirming Op's misconception.

Another example, a reddit thread in askscience somebody wanted to know how scientists knew electrons were the same in the result of quantum tunneling. Someone added that it is because of properties like spin. In reality scientists only experimentally verified/discovered this at the macro level when testing the implications of Shrodingers equation. No scientist actually looked at a single electron to discover quantum tunneling.

The closest thing I can think of is pulling "Pulling something out of your ass" to explain something, but that doesn't seem to quite fit the actual core message that needs to be conveyed, it isn't about a lie, its about the assumption in the first place being wrong, and you just tried to argue/explain something assuming it is true.

There is a word that describes the witless and uncritical acceptance of ideas. It comes from French “fly-swallower” and is:

Gobemouche = A credulous person, especially : one who believes everything he or she hears

Merriam Webster

The gobemouche, having accepted the flawed premises, then goes on to argue on the basis of them.

Regrettably, it is not well known and in explaining it to someone you may, unless you are very careful, make them feel insulted.