They did do the thing that HewlettHackard is describing. Check out the AND gate in the linked article. The input paths are short and use small wires, but also cross the larger paths that normally link the output to ground. If both are active, the paths to ground are interrupted, and the resistor to VCC pulls up the output. So they did make logic gates. In the paper they also demonstrate NOT and OR.
I gather there's a technical sense of "active" that's used in electrical engineering that might not apply here, but to someone like me, with only a tinkerer's knowledge of components, logic gates seem like enough to justify the term in the headline.
I think so? It's section 3.2 of the paper: https://www.tandfonline.com/doi/full/10.1080/17452759.2024.2404157#d1e604
They don't talk that much about current in particular; figure 5 only shows the resistance in the regulated output path as a function of the voltage on the control path, which isn't enough to actually say whether there's specifically current amplification. (Also, the gain would be negative; does that matter?) But they do discuss the fact that the output channel is much wider, which strongly suggests it's able to pass more current (since they mention the resistivity drops as the cross-sectional area of the trace increases). The wider trace is one that wouldn't have the fuse behavior on its own, because the resistivity is too low for it to heat up enough to trigger that at the voltages they're using, but the close proximity of the very thin fusing wire of the control signal is enough to cause a nonlinear resistivity change in the output path as well. I think that means they're using a single voltage for both kinds of path, and that the control current is thus lower than the output current because the resistance on the control path is higher, but I'm not certain. I am not an electrical engineer, just an enthusiastic amateur.