One thing that might help clear some of this up for you is to realize that there are really two issues at play here. Up to this point, the discussion has primarily centered on frequency response. It is important, and does have a notable impact on low level listening. But there is also the power bandwidth to consider as well. Frequency response wise -- which is typically measured at a 1 watt output level -- your Zenith may show some pretty good numbers -- say 30 to maybe even 20 kHz. That would mean that within in a stated level of variation of say +/- 1 or even .5 db, the output of the unit would be consistent over this frequency range at that power output level. But that can all change dramatically as power output is increased.
When a unit is operating at higher output levels, this is where the qualities of the output transformer really come into play. It takes a lot of iron to couple the full power of the amplifier at low frequencies, and well thought out winding patterns to couple all the high frequency power of the amplifier into the secondary load, and not let it all get lost to internal capacitances. The very best transformers can couple the amplifier's full power (with a typical power efficiency of about 80%), with little power loss at the frequency extremes, and therefore little rise in distortion, either. The smaller transformers such as your Zenith has will no doubt give a pleasing sound, and may in fact have a reasonable frequency response. But it is simply not capable of delivering it's full frequency response -- at significant power -- into a secondary load. Therefore, it's power bandwidth is limited.
On the other hand, the output transformers in your Eico have a frequency response to at least 60 kHZ, which will allow it to deliver the amplifier's full power over a range of at least 30 Hz to 15 kHz, which is something the Zenith is simply not capable of. That hardly means to trash the Zenith! It has a part to play in your equipment rotation. However, it also helps to know what it's capabilities are in comparison to the Eico's, so that both can be appreciated for the attributes they offer.
Dave