How do I bind gradient ramp onto each letter?

Unfortunately you cannot "pin" gradients sensibly because most of this stuff assumes a certain default behavior. As the cactus man already advised, split up your work into separate lines or even individual letters if need be. This is one of those things you have to do the hard way like 20 years ago when editable text layers weren't even a thing, much less AE knew what layer styles were and everything had to be its own pre-comp.

Mylenium

Imagine your text is just a solid layer with masks cutting out the individual letters, when you add the gradient ramp it's using the whole of the comp's dimensions.  In the video, the creator uses an expression to get round this, but the whole gradient is still being applied to all the text - so if you have text going over two lines, you'll get the results you're getting.

Instead, take a look at Layer Styles > Gradient Overlay.

Even if you stick with the Linear option, you could use the Edit gradient option to repeat the gradient for the second line.

Alternatively, you may wish to split your text up into individual letters, if that gives you more control.