The article is missing on a couple points. The analysis assumes all Alice's light is emitted radially exactly outward from the center of the black hole. In reality, light is emitted in all directions, and anything emitted at even slightly different angles would get sucked into the black hole. But, Bob might still see it, because he can catch up with it. So when the article says Bob sees Alice cross the horizon when Bob crosses the horizon, it really means that Bob won't see any photons they emitted from inside the BH until Bob crosses into the BH. But similarly, one second prior Bob will be encountering photons Alice emitted roughly 0.99 seconds prior to crossing the horizon, and so on, because those are all getting redshifted too.
It's similar to if you and the car in front of you are accelerating at the same rate, they have a small head start, and they've got someone throwing fastballs at you at 100mph. When you get to the point where you are going 100mph relative to the ground, you'll hit the ball that was thrown from exactly that spot relative to the ground. So, sure, crossing the 100mph barrier implies something interesting mathematically, but it's not something that the observer would particularly notice. The math for GR isn't exactly the same (baseballs won't redshift), and in particular there isn't even a piece of dirt to compare the photon's motion to, as the horizon is just a mathematically-defined "place", but to a first-order approximation, it's the same thing going on with photons and spacetime distortion. It's a continuous function.
There's a bit more to it than can be explained in a comment, but the main thing to know is that (as far as we know) nothing special happens at the horizon if you're falling in.
It's similar to if you and the car in front of you are accelerating at the same rate, they have a small head start, and they've got someone throwing fastballs at you at 100mph. When you get to the point where you are going 100mph relative to the ground, you'll hit the ball that was thrown from exactly that spot relative to the ground. So, sure, crossing the 100mph barrier implies something interesting mathematically, but it's not something that the observer would particularly notice. The math for GR isn't exactly the same (baseballs won't redshift), and in particular there isn't even a piece of dirt to compare the photon's motion to, as the horizon is just a mathematically-defined "place", but to a first-order approximation, it's the same thing going on with photons and spacetime distortion. It's a continuous function.
There's a bit more to it than can be explained in a comment, but the main thing to know is that (as far as we know) nothing special happens at the horizon if you're falling in.