Before it was even put forth by Klein and Gordon,
the very idea of a second-order wave equation,
albeit relatistically invariant, had been abandonned in 1925 by
Schrödinger, who chose to pursue his own first-ordernonrelativistic equation instead.
On the same physical grounds,
Dirac, Heisenberg and Pauli rejected a second-order equation too.
In a letter to Dirac dated 21 December 1926,
Pauli's Hungarian assistant (Johann Kudar)
said that he regarded the relativistic wave equation of
second order with much suspicion.
Soon, Dirac would come up with a first-order equation
whose solutions also satisfy the second-order one.
In so doing, he would invent antimatter...