As a Chinese speaker, Korean very much sounds closeish and Japanese is often similar enough to realize a distant split.
It is important to remember that all branches of a tree grow. China has thousands of dialects all co-evolving and we don't know which the others could have split from and how far either has changed
Korean uses many Chinese loan words (in the same way that English uses Greek loan words) but the sentence structure, pronunciation and grammar are totally different. You would not say English is a dialect or branch of Greek for the same reason.
I speak both Korean, Japanese and am currently studying Chinese. Korean does not sound like Mandarin at all because it is not a tonal language. It does sound similar to Mongolian and Manchurian. That is not surprising because the roots of Koreans (and probably their current language) come from the area north of Korea near Manchuria.
That may have been 500, 1000, or more years ago and each continuously absorbed neighboring language features from trade or due to local social structures.
This is why many of the words sound Chinese (and a particularly old version of Chinese).
Mongolian phonology doesn't sound like any other East Asian language I've heard: https://www.youtube.com/watch?v=OkU9Zd0LQjg
There is a controversial theory that Korean, Mongolian, and Machurian, are all a part of an Altaic superfamily — along with Japanese, Turkish, and a number of others. This was broadly accepted fifty years ago, and less so today.
For example, Chinese has no system of verb conjugation, whereas Korean and Japanese both do, conjugating for the past and present, among other aspects. Chinese also does not have a system of adjective inflection, whereas Korean and Japanese do. Adjectives in these languages inflect to indicate speech level as well as to indicate tense (which is to say, adjectives have a "past form", kind of like verbs).
There are many other structural differences of note.
Beware of arguments based on things we don't know, technically "argument from ignorance". That we don't know which one they might have split from is no reason to believe that this is actually happened! What we don't know didn't happen can't go in place of what we do know did happen.
Another language origin could have been nomadic peoples to the north traveling across the Eurasian steppes, and frankly I'm more inclined to believe that's where Korean and Japanese actually originate from. However, despite the growing study of these peoples that have been neglected in traditional Chinese historiography, they don't leave behind enough archeological evidence for us to prove something like the Altaic language family convincingly, so we're back to null hypothesis and cultural assumptions.
When we look at the grammar of Chinese, Japanese and Korean, it raises a lot of doubts about the possibility of lineal descent -- about the idea that Japanese and Korean are a kind of Chinese. These differences provide positive support for the idea that Chinese bears no especially close relationship to the other two, while discouraging the idea that they are close descendants. Evidence is often like that: supporting what is true while disconfirming what isn't.
My point is that what the "unsupported hypothesis" is depends on your cultural viewpoint. It could be that "not descended from Chinese" is the unsupported hypothesis if you grew up in an environment where most things you experience are descended from Chinese.
Or as another example, if we got into a discussion about whether the sun will rise tomorrow, it's expected that I would have to prove that sun won't rise tomorrow instead of proving that the sun will rise tomorrow. There's no evidence either way until tomorrow arrives. But our baseline assumption is that the sun will rise tomorrow because that's what it's done every prior day. Our null hypothesis are informed by everyday experience, and that's very much a part of our cultural background.
The unsupported hypothesis in argument from ignorance doesn't have to be any particular hypothesis, or a culturally supported hypothesis. Argument from ignorance is a way to frame any hypothesis for which we're not offering positive evidence in a speciously credible way.
P2: "I don't believe this is true" (because my background assumption is that things originate from China)
P1: "What evidence do you have that Korean isn't independent of Chinese"
P2: "None, we have no evidence from this time period."
P1: "Therefore, Korean has an independent origin from Chinese."
P2: "That is argument from ignorance."
P1: "Here are differences between Korean and Chinese."
P2: "I don't believe this is sufficient evidence to enough to revise my doubt about your hypothesis." (because of the strength of my belief in my background assumptions)
What I'm pointing out is that P2's background assumptions and strength of belief in such is inherent to his culture and life experience. This could have easily gone
P1: "Korean has an independent origin from Chinese."
P2: "Okay" (because my background assumption is that things have independent origins.)
Or in an alternate universe
P1: "Korean originates from Chinese."
P2: "Okay" (because my background assumption is that things originate from China)
Where P1 offers some discussion of differences between Korean and Chinese, that's where we start to get into a discussion based on evidence that can be evaluated. If P2 doesn't find it convincing, then we can ask why; this allows P2 to muster evidence which can be evaluated. That's a fine and constructive direction. A finite conversation like that doesn't necessarily end with a solid conclusion and everyone being convinced of the same thing.
What you're pointing out with regards to the perceived strength or weakness of certain assumptions is not what we're talking about with argument from ignorance: argument from ignorance is about the form, not the semantics.