Remember that a flyback transformer is many thousands of turns of hair thin wire, potted in epoxy and sealed up in plastic. Add in years of operation (it's high frequency too!), heat, poweron/off cycles, it starts to degrade the plastic it's potted in, starts to form cracks and the internal wires could slightly shift a bit. The controls are a common place for cracks to form - as they're not perfectly sealed. Any moisture in the air could get into the controls, cause corrosion, arcing, etc. Sometimes you have a flyback with the focus divider failing, such that the picture is real fuzzy and gradually clears up - or starts out dim then gets blindingly bright with retrace lines.
A flyback is a very high stress component, when you think of what all it does. The fact that it works at all is kind of amazing. Anything, when working with high voltage, has to be really well insulated. As soon as the insulation starts to break down you get arcs, which burn up windings and *poof!* bad flyback.
Also, certain flyback designs were just more failure prone than others. For example, the original G07 flybacks, or the K7000 flybacks with the white knobs. They just were marginal designs in the first place - and now they're 20 years old.
The same sort of failures that affect flybacks also affect other kinds of HV parts, like ignition coils in cars. You're pretty much assured to need a new one of those after 100k miles or so.
Fortunately, the most failure prone flybacks are easy to get still. Sure, there are a couple that aren't, but for the most part, you can get replacement parts.
-Ian