Determine the coordinates of the point on the $y$-axis that the force passes

I've recently done a problem where I've been given adiagram with some forces applied to a beam trolley, and where I needed to determine the coordinates of the point on the $$y$$-axis through which the resultant force $$\vec{R}$$ passes.

The problem itself isn't what's interesting in this case, so I won't put out all details and show a whole solution. Rather I've a question regarding this method that'll present to you. So here's my procedure of calculating this coordinate:

1. Calculate the resultant force $$\vec{R}$$

2. Calculate the moment $$\vec{M_0}$$ about the origin $$O$$

3. Solving the equation $$\vec{M_0} = (y \hat{j}) \times \vec{R}$$ for $$y$$

So, my first question is, why does the last step work? I know that the resultant force $$\vec{R}$$ can be moved along it's line of action without affecting the moment about the origin. So, are we essentially moving our resultant vector to a starting point that lies on the $$y$$ - axis, to then calculate for what $$y$$ the force system becomes equivalent to our original one?

I'd be glad if you could present any visual representation that could help me understand this.

Lastly, I wonder why it's even important to care about the $$y$$-intercept in the first place? Are there any applications of this?