sigh - Giving people lessons on the various bias problems that our brain can get us into is great. Basing that on something that is not a falacy is not the way to do it.
Blowing on the cartridge did help, though not for the reason many people usually assume (that it removes dust).
The NES (unlike other cartridge based systems - including the Famicom) had a bizzar angle-loading design.
Usually in a cartridge system, the edge connector of the cartridge should insert straight into the socket, which scrapes off a tiny bit of the metal on each side. This is a good thing - even though it (very slightly) shortes the lifespan of the connector. By inserting the cartriedge, you end up scraping off a bit of the insulating metal-oxides and other junk that builds up on the connector's surface, leaving a direct metal-to-metal connection.
In the NES, however, they decided to use a very unusual Zero Insertion Force connector. Normally, ZIF connectors allow for longer lifespans and are much easier to "connect". This didn't really work properly in the NES, as they had to make the female side of the connector too wide to account for the angle changes when you pushed the cartridge down. Having a bad connection like that causes less metal to be involved, and you tend to get strange electrical effects that can cause more buildup on the connectort in addition to simply starting with a poor-quality connection.
So what does all that have to do with blowing? Your breath adds a small amount of water to the surface of the cartridge.
That tiny bit of water helps the very small amount of movement you get from the ZIF conector to move some of the buildup away,. Even better: (impure) water is conductive, so any marginal connection can sometimes be pushed back into being a working connection for a while.
Now, the water will evaporate away quickly and caused the connector to "rust" much faster, damaging the lifespan of the cartridge significantly). So why did it actually work in practice? Even if it allowed the game to start, after a while the water should evaporate and cause a crash. It turns out one of the BIG problems that prevents a lot of games from starting properly was the 10NES lockout chip. It will force the NES to reset if it didn't like what it saw (the grey screen that blinks at ~1Hz), and it was so badly designed that it was often the first part to start failing when the connector started going bad.
TL;DR - We blew on the board to moisten it slightly to increase the connector's conductivity... as a workaround for the bugs in an early attempt at DRM.
This also describes why a Game Genie helps the NES boot better... while making it worse when you didn't use the Game Genie. The GG had a much wider PCB, that was designed to bypass the ZIF mechanism (you didn't push the GG down like a usual game). This made it act a lot more like a traditional (non-ZIF) edge connector.
The wider PCB ended up fitting a lot tighter, too, and while this gave a better connection... it also bent the socket's pins a little bit. When you removed the GG and tried to use the NES normally, the socket pins were still bent apart slightly. It would eventually result in the socket being unable tyo even touch the cartridge , forcing you tu use the GG permanently.
Oh, and how should you get your games to run, as blowing on it is damaging? Simply use a tiny bit of high-purity isoproponol and some long q-tips. In extreme cases, you can open it up and bend try to bend the pins back slightly. The isoproponol should almost always besufficientt.