Is there a need for this change ?

Because we also don't write type names for for (const auto& x : xs). This is just the extension of a for loop iterator variable imo.

Well, I would say it makes it even more confusing because of the window not being typed. Now if you want to know what types are current and after, you have to go up to the definition of poly. Not super annoying in this case because it is just above, but I really don't see the value in changing a readable typed object to auto. But I know we do have different personal opinions about this topic 😃

official CPP guidelines talk about AAA (almost always use auto), see https://cginternals.github.io/guidelines/articles/almost-always-auto/

I know, but I still don't agree with this paradigm. There are benefits, but according to me they are lesser than the issues it brings, especially on read-expliciteness and the risk of error that it adds of using a wrong type.
I don't really understand that popular languages like Python and JS got a typed-ish version, but on the other side, C++ is moving towards loosely-typed. In some specific cases, auto really saves your day, but the AAA principle looks really counter-intuitive to me.

This seems to fix the symptom rather than the root cause of the issue, or is it the actual proper way of fixing this ?

It has been some time ago that I wrote this code. But from what I recall this was indeed a proper fix since we did not properly account for the end points in the top and bottom convex hull's.

Yes, just re-read the code. This is indeed the proper fix. We're doing the following things.

1. sort points based on it's x-coord (from low to high. If x-coord is the same then order is defined on y-coord
2. maintain a invariant of the convex hull, $P^{\text{convex}}$, then for each point $p$ execute the following sub routine
   a. check if adding the current point invalidates any previous point $P^{\text{convex}}_{|P| - 1}$. as long as the $P^{\text{convex}}_{|P| - 1}$ is no longer convex given point $p$ and $P^{\text{convex}}_{|P| - 2}$ remove $P^{\text{convex}}_{|P| - 1}$. Do this until the $P^{\text{convex}}_{|P| - 1}$ is located on the convex hull invariant.
   b. add the point $p$ to $P^{\text{convex}}$
3. Reverse the points and perform the same procedure defined in 2

The issue here is that we're always adding the current point in step 2.b. However, it could be that the last point(s) are not on the convex hull (due to the stable sort sorting first on x then on y, so either for the forward pass or backward pass it could be that the last inserted point is not on the convex hull). This code is to fix this oversight where keep removing the last point(s) untill all points are convex.

Well, I would say it makes it even more confusing because of the window not being typed. Now if you want to know what types are current and after, you have to go up to the definition of poly. Not super annoying in this case because it is just above, but I really don't see the value in changing a readable typed object to auto. But I know we do have different personal opinions about this topic 😃