Are you sure that you can only root Samsung S20 Ultra with computer? Does that only apply to S20 ultra or all android? Im not saying you guys are wrong, Is just that I read so many things online if you google root without computer a milion results and guides come out. Sorry if this is stupid question.
The problem is that the internet is a great repository of information, but it's also a great repository of out of date information, or plain clickbait, deliberate misinformation or information that the person posting it hasn't checked. But any site that tells you that King Root or Kingo Root will root an s20 Ultra is pushing misinformation. That may just be because to generate clicks they automatically recycle the same stuff they wrote years ago with newer device names, without caring whether it's right or not - I can only speculate as to why they would post such stuff. But those one-click-root apps have not worked for years.
So yes, prior to Android 6 all of this stuff worked. You may find some more recent devices where such exploits still work - Chinese "fake" phones often use old Android versions that have been reprogrammed to pretend to be newer (seriously!), so they would still have the necessary vulnerabilities. But an s20 Ultra? No chance. In fact any even reasonably modern android will need a computer. Even with an older device, if you still hope to be able to use things like banking, payment apps or many media apps, I would avoid a non-computer method: the root methods that are best at hiding root very definitely rely on a computer (indeed if you are thinking of rooting I would first check whether any individual apps that you care about still work after root: don't rely on some statement like "this Magisk module hides root", check whether it allows that specific app to run).
One thing you've not said is what type of s20 Ultra you have? If it's a North American Snapdragon-based model then the question isn't whether a non-computer method can work but whether it can be rooted at all: recent Snapdragon-based Samsungs are very hard indeed to root, in many cases impossible. If it's a Exynos-based model (as sold in much of the world) then you have a better chance, e.g. my s21 has an official bootloader unlock method, which a Snapdragon-based one would not have.