Disabling 10NES Chip
Solution 1:
The 10NES chip can be effectively disabled by cutting pin 4 on the chip. (The pins are numbered starting with the black dot as 1, and then continuing down the same side of the chip as the dot).
I did this to my old NES and it worked fine. However, any hardware modification can be somewhat risky, so your mileage may vary.
There's a picture of the chip and the pin to cut on this wiki or there's also this YouTube video that goes step by step.
You can tell if the 10NES chip is disabled because the 10NES chip will reset the console over and over again if a valid cartridge is not inserted. If your power light doesn't blink constantly anymore when the console is empty, then you did it right :)
As far as I'm aware, there's no downside to removing the 10NES functionality. In fact, later NES models that were compatible with the same cartridges did not include it.
For a more in-depth look at the 10NES chip, and a perhaps more in-depth way of disabling it (by routing pin 4 to ground instead), there's this guide.
However, some NES reliability issues are related to the funky 72-pin connector on the NES-001, rather than the 10NES chip directly. You may also have to clean, refurbish, or replace this connector to improve the console reliability, depending on the particular issues your hardware has.