That "speaker amplifier" has probably upwards of 60 dB of open loop gain and what looks like multi-feedback topo (cathode of pentode grounded through secondary of OT). So as tube amps go, it's actually a pretty good circuit. But distortion will depend somewhat on load. They probably designed it for 1% at nominal output level. Something like that. So you if back-off the load a little (not too much or the frequency response could start to get wonky), you could bring the distortion down a little. So yes, it could very well improve the quality of the signal. If it did not measurably improve the quality of the signal, I would wager that that would only be because the signal quality was already good enough that lighter load just doesn't make that much of a difference.
But I personally would not care about a little distortion. In fact, I would probably crank the thing into 1%+. Isn't that half the reason for using a machine like this anyway? If it were not for the fact that the capstan is probably spinning when you turn the machine on, I would probably run signal through it monitoring input just for the sweet tone of a band-passed single ended pentode output.