Do we actually take random line in first step of linear regression? 
This is the screenshot I took from a video on linear regression made by Luis Serrano. He explained linear regression step by step (scratch version). The first step was to start with a random line.
The question is do we actually draw a random line, or instead do we perform some calculation like taking an average of y values and initially draw a line. Because if we take any random line it might not fall near any points at all. Maybe it will fall on the 3rd quadrant of the coordinate system where there are no points in this case.

That example is definitely NOT the way linear regression is typically done, but I suppose it is an algorithm to find a regression line.  As other answers have correctly stated, there is a closed form solution for finding the Least Squares Regression equation for a set of points.
That being said, what's being shown in the snippet is a method for algorithmically finding a line that gets close to the points by trial and error (i.e. iterations).
As a simple analogy to show the difference between a closed form solution and an algorithm: if I were to give you a mathematical equation, say $10 = 2x+4$, and asked you to solve for $x$, we know that you can solve this exactly using algebra.
$10 = 2x+4$
$\implies 2x=6$
$\implies x=3$ ** Exact solution **
Alternatively, an algorithmic approach to this could be used to solve this same equation by guessing a solution (e.g. start with a random guess: $x=0$) and systematically adjusting $x$ until your condition (statement of equality) is met, or approximately met.
$x = 0 \implies 10=4$ ** too low, adjust up **
$x = 1 \implies 10=6$ ** too low, adjust up **
$x = 2 \implies 10=8$ ** too low, adjust up **
$x = 3 \implies 10=10$ ** condition met, stop **
As this crude example shows, algorithms can sometimes approximate the answers returned by closed form solutions, but this isn't guaranteed to happen for all types of equations.
Personally, I don't find the snippet in the question to be pedagogically helpful to showing how regression lines work, and I think there are better examples of how algorithms can be used to find approximate solutions to mathematical equations.
