Not necessarily. An early-16th century arquebus is terrible in a one-on-one contest with a bow. Bow's faster, quieter, sometimes even better for armor penetration. The key is that the gun can be used with a few hours of training tops, and used competently in weeks, whereas a competent archer takes between several months for a really simple bow to a decade or more for a longbow or some recurves, so the replacement rate of trained ranged troops is really low.
Guns and bows also have better effective range than atlatls or thrown javelins, that's extremely useful.
A gun has longer lethal range and comparable effective range, incomparably superior stopping power born of the combination of six times the energy for beating armour and something like twelve times the lethality against unarmoured foes (it's a ball of soft lead that expands when it hits bone, just picture it), can be used from prone or from cover, and doesn't care about leaves or branches that get in the way. And this incredible powerful weapon, while heavy, doesn't require the kind of athleticism a bow does.
Also using one is not easy: loading a 16th c. gun is a 16-step process and if you screw up, you'll blow your hand off. And if you really screw up, you'll light up your whole bandolier and turn yourself into a fireworks show with a chance of meaty salsa. Maintaining it isn't trivial either. A good gun marksman was therefore always prized. It's much easier to aim with a gun, of course. You only have to really worry about not aiming too high and about the recoil: so for comparable amount of training, a gun is also more accurate than the bow.
A bow is more silent and you can shoot with it a good deal faster, true. But the firearm is a hands-down winner in every other aspect, as borne out by history and specifically the very native American nations that we were discussing earlier. They were all very eager to adopt it and it changed everything. It broke generations-old deadlocks and caused horrific slaughters and mass migrations. But no matter how much the American nations loved the gun, they couldn't reproduce the technology end to end.
EDIT: Okay, here's my thoughts on it: the Aztecs have the best chance of survival if the Spanish don't have monopoly in Mexico. If there's no Tordesillas (earlier POD than Cortez, I know), perhaps France and Portugal would get in on it
right away and create that dynamic similar to east coast of US/Canada where the Europeans constantly supply their allies with whatever it is they need to keep fighting their native and European enemies. Can have England and Netherlands and whoever else join in after, too.
It's also an under-explored scenario, in my estimation.