A treatise on the theory that you can extend your bottom time with nitrox OR dive more safely with nitrrox, but not both.
It is obvious that you can extend your bottom time with nitrox.
The theory behind the safety benefit always says to dive with nitrox but to air limits. For example, let's say I did 2 dives with a 1 hour surface interval, with the first one to 110 feet and the second one to 60 feet. I use nitrox 32 for the first one and nitrox 36 for the second, but I dive both using the PADI air tables. Diving on air, my first dive time limit would be 16 minutes, and I would be in pressure group M. After one hour, I would be in pressure group D. My maximum bottom time for the second dive would be 39 minutes, after which I would be in pressure group W. If I did those dives for the same lengths of time using those nitrox mixes, I would end the first dive in pressure group K. After 1 hour, I would be in pressure group C. After the second dive, I would be in pressure group O, which is quite a bit farther from limits than the air dives.
The reason people say that you can't get more bottom time AND more safety assumes that the diver on nitrox will dive to the limits of the NDL nitrox, too. If that were true, there would indeed be no safety advantage. But there is no requirement that people dive to those limits on nitrox; in fact, I would say they usually don't.
Let's do the above dives using the nitrox but doing the dives for the lengths of time we see on typical single tank dives.
The NDL for the first dive on EANx 32 is 25 minutes, but most people don't have the breathing capacity to last that long on a single tank dive, so let's say the diver ascends after 20 minutes--adding 25% more bottom time compared to the air limits. This puts the diver in pressure group M--the same as the air diver. After 1 hour, the diver would be in pressure group D, again the same as the air diver. For the second dive, the nitrox diver has a maximum time of 92 minutes, but he isn't going to last that long on a single tank at that depth. He ascends instead after 50 minutes.He is now in pressure group R.
In summary, the air diver got a total of 16 + 39 = 55 minutes of bottom time, and ended in pressure group W. The nitrox diver got a total of 20 + 50 = 70 minutes, more than 25% more bottom time. That diver ended in pressure group R. He would have had to stay in the water another 25 minutes on the second dive to get to the same level as the air diver.
It seems to me you can indeed get both extended bottom times and greater safety.