Yes, you're wrong. Sorry, that didn't mean to sound blunt, but you asked. Without a tail light, the current through R1/R2 goes to zero so V=IR goes to zero, ie the cap fully discharges. This creates a poweroff detection
With a tail circuit, there is current through both BR and R1/R2 and V=IR is NOT zero. It's higher than zero. So there's a voltage across R2. If that voltage is too high, OTSM doesn't work. In fact it always means voltage will drop at a slower rate and OTSM won't trigger quite as fast, but that shouldn't be a big deal so long as it gets low enough pretty fast.
This wouldn't work at all if not for the fact that the diode itself has what, 2.1V. So for a full battery the total voltage across all other resistors when off is only 4.1-2.1 = 2.0V. That helps. We only need to get to, er, about let's say 0.8V. (technially 0.3*Vcc, but Vcc has also dropped a little by then, and you want to dive below spec a bit to get there faster, 0.8 is a good ballpark for a full battery.
So you can actually calculate all this, but it's tricky because of the non-linear diode. If you just assume a fixed diode Vf then it gets easier, but you probably want to target a certain diode current to get that. Then use Vbatt-Vf as the voltage over the resistor network, then work out the current in Rled, then iterate, or course use algebra, which I've been meaning to do. If someone can toss out a good number for a target Vf and diode current that match, it will help me make a start on that algebra.
There's a reason I never wrote this chapter. It requires extra thinking.
Anyway, the simple version is you need either lower BR or lower R1/R2 or both and/or the LED resistor higher. If he was close before, then just trying something significantly lower should be good. And adding a 1k BR does at least significantly lower the C1 parallel resistance.