Those are two different scenarios.
The first was awesome and exhilarating.
But let's be honest. It was a shotgun with a limited round capacity that had been spent anyway. I think the maximum is around 6-7
Lee's working with what appears to be a pistol which has around 17 round capacity.
In each encounter he definitely doesn't take that many shots. So we can assume he reloads in between encounters. There are many moments when the scene isn't focused on Lee and when the story moves forward. So it's entirely plausible.
Anyways. I sort of agree with you somewhat. It would definitely be interesting and pump up the tension. But do you really want to be searching for ammo after every encounter instead of continuing the narrative. Which is what adventure games are all about. Cause if that's the case after every encounter Lee would have to go on a mini-quest to find more 9mm rounds. Which would completely derail the story and ultimately make it more difficult to construct a solid narrative.