In the beginnings of quantum theory, people were looking at the K-G and the Dirac equation as equations for wave functions (or at least something similar that would give them a probability density like the non-relativistic wavefunctions did) - the notion of a "quantum field" did not yet exist.
As an equation for such (generalized) wavefunctions, the K-G equation is rather obvious "nonsense" - not only does it have "negative energy solutions", but its solutions also produce negative probability densities (see e.g. this answer by gented). So negative energy solutions to the K-G equations weren't really hinting at antiparticles, since everyone knew the solutions to the K-G equation didn't produce meaningful quantum states anyway.
In contrast, the Dirac equation as a first-order equation gives solutions with positive probability densities, so its solutions can be interpreted as defining quantum states, and so its negative energy solutions "suggest" antiparticles.