Actually you can. The way you balance it is by making the enemies use mostly low end weapons at the start then gradually increasing weapon power the farther you go in to the game either measured by distance, time, or both. This way you can control what the player gains access too. JA2 did this to a degree in addition to not having 100% loot drop.
One draw back is you fight a lot of pistol/SMG enemies and it makes progress feel slow. This gives a feeling of forced, gated progression knowing that you'll never get a really cool gun from a drop for a while. Another is that enemy diversity/power options are lacking as you can't give them better guns but have to rely on numbers, positioning, and map layout to challenge the player. That might sound good on paper but it actually limits the strength of your enemies quite a lot forcing them to only have a chance via cheesy tactics.
As for ammo shortages specifically, as I suggested before maybe they could just increase the ammo drop rate and not the gun drop rate which would help alleviate the situation. Also more ammo crafting locations, including adding one on the starting island, would help.