It does; once the petals fall off, the flower is what becomes the fruit. If you dead-head them, the rose won't be able to put energy into growing the fruit, and will make new flowers again in the hope of making a fruit that'll survive. But you do have to get all (or almost all) the heads off, or it'll push energy into the ones it's got left.
In the case of roses, the fruit is rose hips. One day I'll learn what can be done with them, I had a great aunt many, many years ago who picked the hips of wild roses to make syrup but not sure if cultivated ones can do that.
rose hips are very medicinal! they have high amounts of vitamin C and are great for syrups like your great aunt would make as well as in a topical oil for the skin.
And thank you for the advice about dead heading the roses. I will give that a try to get another round of roses on a couple of the more prominent bushes.