If the geodesic is infinitesimally close to a null curve, shouldn't it also have zero path length?
This is a similar paradox to the "$\pi = 4$ paradox". Take a semi-circle of radius 1. We know it has length $\pi$. But we can approximate it by superposing the circle on a grid of pixels and tracing it out in a jagged way. As you make the grid finer, you can keep approximating the circle with arbitrarily nearby circles, each of which has a total length of 4. Thus, $\pi = 4$?
The important mathematical takeaway is that the limit of a function of a sequence is not always equal to the function of the limit of the sequence. Here, the sequence is the sequence of ever-closer arbitrarily nearby circles, which converges to the circle. The function takes an element of that sequence and spits out its length. So the former is 4, the latter is $\pi$.
Physically, the important takeaway is to remember to what order a quantity contributes to a sum. 586
As the size of the elements you're cutting up your path into gets smaller, does the thing you're using to approximate the actual length of the small path segment really tend to the true path segment length? Turns out, "angle" information contributes to $O(h)$ to the actual length of the small path segment, so to $O(1)$ to the final integral. Whoops!
Why does this imply maximising proper time?
Well, what's the condition for the path to minimise the proper time? It means that no matter what small disturbance $ \epsilon(\tau) $ you pick around that extremal path, the path that results from adding that disturbance to the path produces a new path that has longer proper time. So the strategy, if you want to show that the path doesn't minimise the proper time, is to find a nearby path that has shorter proper time.
Here <PERSON> suggests one possible such path, built up from null segments. Is this a convincing argument? Well, I haven't read the text, but I do have some possible objections. First of all, showing it is not a minimum is not the same as showing it is a maximum. Could it not be some kind of "critical point" in the path-space? Maybe someone more familiar with the mathematical technicalities can show this isn't the case.
I would also object that the non-differentiabilities of the jagged path could conceivably contribute non-negligibly to the final length. I think you could fix that part of the argument by "smoothing out" those bends and show that the contribution is bounded by the size of those smoothed-out turns. However, there would be some technicalities to do with which limit is taken first: the cutting up of the path into many kinks, and the "pinching out" of the bends into kinks. I've also got in mind the twin paradox, in which this exact issue can come up, and you have to think carefully about why the contribution to the accelerating twin's proper time when they're turning can be neglected as the turnaround time goes to zero. 795