Flame Breathing could actually increase the power of Fire Release!
The Hokage version of Flame Breathing could also enhance physical abilities, but that was not its primary purpose. Its main purpose was to strengthen Fire Release.
Once Flame Breathing was learned at the beginner level, one could use Total Concentration Breathing to instantly speed up Fire Release Chakra Nature Transformation and enhance the power of flames.
After becoming proficient in Flame Breathing, one could enter a constant state, making both the accelerated Nature Transformation and enhanced flame power permanent, while also gaining greater explosive strength.
The Marked State after mastery was even more terrifying. It allowed one to instantly convert all Chakra into flames without any hindrance. All Fire Release Ninjutsu could be cast immediately without hand seals, greatly increasing the power of the flames.
After the system's extensive modifications, Flame Breathing no longer had much to do with the one from Demon Slayer. It required no sword at all and could be called a Fire Release Chakra Mode.
There were many kinds of Chakra Modes in the Hokage world. The more common and practical ones included Sage Mode. Jinchuriki also had the chance to master Tailed Beast Chakra Mode, and there was Kumogakure's Lightning Release Chakra Mode.
Mu Yue even felt that Susanoo could count as a Chakra Mode as well, a Chakra Mode exclusive to the Mangekyo Sharingan.
Some Chakra Modes could even be stacked. Mu Yue did not dare imagine how powerful Fire Release would become if Sage Mode and Flame Breathing's Marked State were used together.
Although the reward was not the Geppo he had imagined, Mu Yue was very satisfied.
The localized Flame Breathing was extremely effective. It could both enhance his own strength and suit his disciples' training.
Shisui and Obito primarily used Fire