I am usually a big fan of thanissaro's, but he loses me here.
I would say the two methods are in fact different but still clearly aimed in the same direction. Let's take a practical example. I feel very angry because I had the perception that someone thought I was stupid. In my mind I am replaying the scenario, there is mental vedana and tension in my chest. If I recognized this and I was practicing bhante style, I would simply be aware of the thoughts and feelings, the thoughts would stop for a moment as I stopped fueling them, but I would forget to just engage in bare attention and would instead start recreating the thoughts. I would then again remember, through noticing that it is suffering to trap myself in that thought-world and I would go back to just attention. The tensions and mental vedana would likely persist as subtle thoughts are still occurring, but slowly it would die out, I would move towards just experiencing the world without creating these unfortunate mindstates. Over time, doing this again and again, I would slowly recognize more and more that the pure awareness is better, and furthermore the pure awareness would become more and more pure until eventually I'd become capable of truly desiring nothing. Fabrications are intentional and impermanent so if you stop intentionally creating (some of them) there will be less of them, and if you keep working at this you will get better at the method (practice makes perfect) and you will be able to fabricate none of them.
With thanissaro's method, I would, upon recognizing the unfortunate mindstate begin to actively combat it, as he suggests. I would alter my pattern of breathing, working through the tension and mental vedana intentionally with visualization and directing breath energy. I would re-formulate my narratives about the scenario - were my rights really violated? does that person thinking of me as stupid really harm me? As such I would eventually exert a fabrication to still the unskillful fabrication. Over time I would do this a lot, lose my desire for negative fabrication and only fabricate jhanas, I would then apply essentially the method above to stop fabricating entirely/desiring nothing.
Is one of these better? I'd have to say that from the pali canon the buddha's stance would be that the latter was better, but the former would still work, clearly in non-theravadan schools such as soto zen or dzogchen the former is relied on exclusively or almost exclusively.
I look forward to your criticisms - do you agree that both methods would work in roughly the way I suggested?