Flyback is shot. Repairman says part is no longer made. True??

Well, I thought I could look up a replacement if you really wanted one. To answer your question, the original part is probably no longer available from the manufacturer. This is especially true for a small TV where the replacement cost for the whole set is pretty low. But if the set was a popular model or used the same chassis as enough other models, a replacement flyback might be available. So the tech was probably right.