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.

To be clear, there's a closed form solution for linear regression that is almost always used to find the fit, so there's no need for a "guess" to start with at all. This example is more of illustrative example of how Stochastic Algorithms work rather than how to best fit a linear regression model.
However, linear regression really is the exception to the rule in this case. For fitting most models, we do not have a closed form solution and we do need to start with an initial set of parameters and then iteratively improve them.
In such cases, it is often the case that choosing a good starting point, as you have suggested, will help the algorithm to converge faster. For some problems, choosing a good starting point is crucial for acceptable performance (both in terms of speed of convergence and probability that algorithm will converge to an acceptable answer), while for other model/algorithm combinations, the improvement may be so minor that it is not worth the extra effort to find a good starting values and initializing with random values is fine.
