How did people infer that the strong interactions have a large coupling prior to the discovery of QCD? How did people in the 1960's determine that the strong interactions were governed by a theory with a large coupling constant if they were not aware of the underlying theory, QCD, which was only discovered in the 1970's? What is the experimental hallmark of a strongly interacting system? I have read that there were severe doubts during this period over whether QFT even applied to the strong interactions---did one need to presume a QFT framework in order to make the statement that the coupling was large?
 A: Before the discovery of QCD and the quark model, particle physics consisted of interactions of electrons, protons, neutrons, and all those mesons, called mesons because they were in mass values intermediate between the electron and the proton/neutron. That is how the muon came to be called a meson. Some mesons were used as the exchanged particles in effective field theories to explain cross sections and lifetimes. Pion exchanges and rho meson exchanges were used , 

In physics, vector meson dominance (VMD) was a model developed by J. J. Sakurai1 in the 1960s before the introduction of quantum chromodynamics to describe interactions between energetic photons and hadronic matter.

to model the data, in a quantum field theory framework with Feynman diagrams etc.
At that time the weak interaction was described by the four fermi vertex also.
So people knew that there were weak, electromagnetic, and strong forces, the strong force postulated for the proton and neutron, the nuclei and the strong resonances when scattering pions and kaons on protons, for example.The resonances were wide giving short lifetimes, differing from the long lifetimes of electromagnetic interactions and weak ones.
It was the classification of the numerous strong  resonances discovered in particle  deep inelastic scatterings that gave rise to the quark model, i.e. a definite substructure to the strongly interacting particles organized into group representations that gave rise to QCD.
Before the quark model dominated the scene there was the parton model which Feynman proposed, and he was a champion of, long after the rest of us were convinced that QCD and the quark model should be considered the standard model.

he parton model was proposed by Richard Feynman in 1969 as a way to analyze high-energy hadron collisions.2 Any hadron (for example, a proton) can be considered a composition of a number of point-like constituents, termed "partons". The parton model was immediately applied to electron-proton deep inelastic scattering by Bjorken and Paschos.

The discovery of jets and particularly of gluon jets  cleared up the issue, i.e. that the hadrons had a definite hard core giving rise to deep inelastic scattering of the Rutherford scattering type and not a soup of partons.
A: The strong interaction is only strong at small energy. At high energy, on the contrary, we have asymptotic freedom. This was discovered by the study of deep-inelastic scattering (DIS), i.e. electron-proton collisions where the proton is destroyed. It was actually a big surprise at the end of the 60's that the total cross-section of DIS exhibit a property known since then as Bjorken scaling, because a simple explanation of this result can be obtained by assuming that the proton is made of non-interacting constituents, which were named partons by Feynman. This discovery was a turning point because it meant that if we had a QFT for the strong interaction, then perturbation theory could work in high-energy experiment. Most theorists highly doubted that to hold, precisely because the strong interaction looked too strong for perturbation theory to work as in the only other QFT they knew, QED, for which the coupling constant $\alpha$ is small enough that any observable can be computed as a series in $\alpha$ using perturbation theory. 
A: We knew the (old) strong force required large coupling because there are nucleii stable against electrostatic repulsion of their protons.  Independent of theory, whatever held nucleii together had to be strong enough to overcome electrostatic repulsion.  (A proton about 1 fm away from another produces a repulsive force of magnitude about 100 N.  Which is rather a lot.)  Nevertheless, there are stable nucleii.
(That the nucleus was so small came from Rutherford's experiments in 1909.  The same experiments showed that electrostatic repulsion continued to be large at very small impact parameters, indicated by the presence of very large scattering angles.  Thus, the need for a large coupling constant has roots in Rutherford's experiments.)
The strong interaction has been two inequivalent things in my lifetime.  That the first one involved a large coupling was phenomenologically driven -- there are stable nucleii.  That the second one involved a large coupling was driven by the fact that the first one was a miniscule consequence of the second.  (Since I'm throwing around one number, let's throw out another:  the strong interaction has magnitude around 10 kN.  As noted above, the nuclear force is about 1% of this.)
