Q.pyracantha bushes not having berries
A row of firethorn bushes is planted along the northeast side of our garage. There are 7 or 8 bushes crammed into this space and they all seem to be growing into one another. Two bushes on one end of this row have orange berries on them; the rest just leaves. Would it be possible to remove every other bush to get some air in there without damaging root systems of the ones I want to keep? Would the thinning procedure encourage remaining bushes to bear fruit? Is there something else I can do to change these bushes from an eyesore to an asset?

Pruning and some soil corrections may be all you need.
Pyracantha are often planted and grown as a hedge so if this is the reason for you plantings you may not want the look of spaces in-between the bushes.
You likely would do some damage to plants in attempts to remove some but if you do attempt this wait until the they are dormant.
I would start with a soil test. Your lack of berries is likely from to much Nitrogen in the soil.
You can adjust this with an application of Phosphorous rich fertilizer like Bone Meal.
Here are some links for you.
https://www.gardeningknowhow.com/ornamental/shrubs/firethorn/care-of-firethorn-bush.htm
https://www.gardeningknowhow.com/gardengroup/pointy-plants/
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/bone-meal-fertilizer.htm