Sunflowers are an annual. Annuals have a short lifespan so their primary goal is to make SEED (babies). Any annual will bust their little sepals to make seed before the season ends. Thwarting them making seed by cutting off the flowers makes the entire plant get bigger and more vigorous and put out more flowers. Sunflowers aren't the floriferous type...if you've pinched them back soon enough, possibly get another flower or two but sunflowers, once they've got a mature flower that will be it. They last a long time as they are making seed and need to attract the pollinators to pollinate all those seeds. Once they are pollinated, the flower is no longer necessary for attraction.
Next year, plant more sunflowers and if you plant them 2 weeks apart you'll get successive and continuous flowers. Cut 'em off and use indoors, this will entice the plant to produce more. But sunflowers are seriously a one flower event...even if branched like some of the smaller varieties are...they get 4 or 5 flowers...still plant seeds every two weeks and you'll get more flowers.
Sunflowers are the TOP of the evolution for dicots. Orchids are the top of evolution for monocots. Sunflowers know how to produce lots of seed and rise above other plants to compete for sunlight. Because they are so big there isn't much chance of budding after the main crop is established. One flower is all they need to make to get a huge bunch of seeds/babies. Other annuals depend on the numbers of flowers. The best thing you can do other than watering and a little fertilizer is to remove the lower, yellowing leaves as soon as possible. This reallocates energy to the flowers and producing leaves. And definitely have other fun things to bloom at the same time or different seasons/times. Sunflowers do last a long time for most flowers and they make anyone smile!