Why do people hand wind flyback transformers?
Flyback transformers require a very small air (or other dielectric) gap within the magnetic core structure. This helps prevent core saturation but lowers inductance. Most transformers that you're familiar with aren't the flyback type so they would saturate early and fail to function well as a flyback so you can't just swap in a random transformer from some old, no longer used electronics usually. Between difficulty finding the exact right type of transformer for your application, cost, availability, and ease of finding magnet wire, it's not terribly surprising that many choose to make their own for one-offs.

Lastly, although using old transformers from other electronics may work, it likely won't work well because there's engineering tradeoffs that are made that probably aren't anywhere near ideal for your application. In total, not only are you looking for the correct ratio of turns, you want wire thickness that will keep your resistive losses to a minimum, inductance that's suited to your circuit, and in this case, an airgap in the magnetic core.

See here for more info on why flyback transformers need the special air-gap.