Focal length is exactly magnification. That is the only reason we use telephoto, to magnify the subject size.
This is simply incorrect. It's not magnification, and that's not the only (or even a very common) reason to use telephoto.
You primarily use telephoto not to magnify, but rather
to get further away from the subject. Or conversely, to take photos of things you can't get close to.
If a human being is the subject of your photo takes up exactly one half the height of your frame in focus, then your magnification is the same, no matter which focal length you take that photo with. Longer focal lengths are useful though, because if you take it with 15mm lens, you have the same magnification, but the person will look weird since you're closer than you normally are to people. With 100mm lens, same magnification, it looks more normal, because you're 10-15 feet away.
Distance away from a subject =/= magnification.
Magnification = how big a thing is on your sensor compared to how big it is in real life.
If you are 15 miles away from a ladybug, and take a photo of it that fills half of your frame using a lens the size of an apartment building, then your lens still has
LESS magnification than if you are a foot away from the ladybug and take a photo where it fills the entire frame.
The problem here is just that you're using a colloquial definition of magnification where it is actually a pretty specific technical term in photography (and biology, etc. etc. same technical definition) where focal length is just one of two variables you need to solve the equation.
A number of years back, I shot 1:1 magnification images using my 60mm Micro-Nikkor and my Sigma 180 f/3.5 EX Macro...I got what appeared to me to be the same depth of field with the 180mm lens as I did with the 60mm, despite the 180mm lens being 3x farther away...it kind of made an impression on me...
Well yeah, the chart above seems to show that it becomes essentially indistinguishable as low as 0.05 magnification. So if you're shooting 1:1, the DOF will long past being affected, probably completely indistinguishable / sub-pixel differences kind of thing.
At 0.01 magnification and lower, though, it's a major difference. 0.01 = things about 70 inches tall filling the vertical on a full frame camera. In other words, something along the lines of a seated human portrait with plenty of space above the head.