The motivation is better than most, trying to treat infinity as just another case instead of an exceptional one, but, to me at least, it still feels like a "hack" to put in a third dimension and then insist on dividing by it to reproject whenever you want to do anything useful.

I'm a little used to the trick but it'd be nice to give some deeper motivation of why the trick works in the first place and why this particular representation.

If you want to study how a polynomial P(X,Y) behaves at infinity, you can't set eg X,Y=infinity, but you can take P(X/W,Y/W) when W goes to zero. Now clear denominators by multiplying by an appropriate power of W and you get a homogeneous polynomial in X,Y,W. You can now set W=0 directly. Scaling a solution cX,cY,cW doesn't change the original solution since (cX)/(cW) = X/W, etc.

I guess the important part is clearing denominators turns problems with division into linear problems.

It's a nice extension of the plane to get some results. For example https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem (Gemini recommends me this video, but I didn't see it https://www.youtube.com/watch?v=5OYLnMyesX4&t ). The idea is that in the complex protective plane, two lines intersect in exactly a point, and two circles intersect in exactly 4 points (yes 4). It's very easy to count intersections there, so after you counted them there you can try to count how many of them are in the usual plane and how many are "fake" intersections in the extension.

There are many similar results, but this is one of the most easy to understand.

Lemme try. Have you ever seen a railroad vanishing in the horizon? You know the rails never meet and yet, they definitely meet in infinity before your very eyes. That's because you see a 2D projection of a 3D space. In 3D, you'll never find the intersection. It gives one of the best intuitions on how to work with points in infinity.