Yes, they do. Modern industrial farming relies on the exact same fossil fertilizer inputs. When it comes to recycling phosphate etc., the rival to hydroponics and conventional farming is regenerative, soil-based agriculture. The latter relies on composting and manuring to turn organic waste back into nutrient-rich soil. Among other functions, microbes in the soil and compost heaps break waste products like urea down into bioavailable forms like ammonium. Hydroponics generally doesn't involve composting and is by definition soil-free, so the plant's rhizosphere is more or less sterile:

https://en.wikipedia.org/wiki/Rhizosphere

To farm plants without soil, they must be supplied with a water-based nutrient solution. The solution contains a precise mixture of all chemical substances a particular plant needs to grow. Yield efficiencies can be realized by carefully tailoring this mixture to each plant. The problem is where to source the ingredients once extracting them from mineral deposits turns uneconomical.

Compost- and manure-amended soil theoretically has everything plants need, but that's not very useful for soil-free farming. The precise ratio of nutrients in a given patch of soil is also outside of the farmer's control. Perhaps the individual nutrients can be leached, distilled, or otherwise chemically separated from compost and biological waste products to enable their reuse in finely-mixed solutions? I don't know. Maybe someone with more knowledge can answer.