But the voltmeter was connected only across the lamp, the point being to determine how an unmodified filament behaves when the voltage across it varies in small increments. And for that purpose, as long as the measurements are taken at equilibrium, it doesn't matter whether R1 is temperature dependent or not.
So we know that small changes in the voltage across a lamp result in proportionally smaller, but measurable, changes in current. Now replace R1 (my variable resistor) with a second lamp. Apply the appropriate voltage to the string and note the current. Thin the filament in the second lamp with a laser, or a genie with an angle grinder. As we agreed before, the network will reach a new equilibrium, with the voltage divided according to the new ratio of the filament resistances, such that the voltage across the unmolested lamp is slightly higher than it was before. Consequently, the current in the circuit will have increased a small, but detectable amount, and so has the sum of the power consumed by the two lamps.
Which is contrary to what was reported in the article. In other words, a localized thinning of the filament can't explain an increase in brightness without an increase in power.