Why piston aircraft engines never became completely obsolete

Who first recognized that piston aircraft engines were obsolete?

No one conclusively did, because piston engines remain useful and continue to be built for specialist aircraft.

Piston engines never became obsolete at a single moment. New examples continued to enter service after jets dominated high-performance military aviation. The technology moved toward smaller manufacturers and specialist aircraft rather than disappearing. Missions that do not demand extreme power or speed can still benefit from a piston engine's established operating characteristics. Rugged operation from rough or nonexistent landing strips can also favour piston power even when gas turbines scale better to high output. The useful question is where the technology remains suitable.