Is this way of teaching how to solve equations dangerous somehow?

The worst thing to teach is that mathematics is a series of recipes to be blindly followed. Mathematics should be about ideas. The reason "walking from one side to the other" works is precisely that it preserves the solutions to the equation. Teaching the mechanics of solving equations without even mentioning that reason should be a criminal offense.

The textbook from which I learned basic algebra used exactly that idea. Further, I think numbers walking to the other side teaches use without understanding and should be avoided.

I think the way described in Morris Kline's book is better.

It teaches the basic principle - the meaning of the equal sign =.

In your first example, you add (-8) to both left hand side and right hand side when going from (1) to (2). In other words, you are solving the equation (to find the value of $x$) by keeping the left hand side equal to the right hand side.

I don't see any basic principle behind walk from one side to another.

Convergence is a much more advanced concept than a simple equation. It involves more than the basic principle of equality. If one fails to comprehend what equality means, I don't see how he can understand convergence.

So, yes. It is dangerous to teach students how to solve equations without introducing them the basic principle behind it.