XLR out on a tuner?

What Denon did here is common practice in the industry, even now, and even in much more expensive components with high end pretensions.  Don't hate them for it.  In fact, some equipment has XLR inputs or outputs that are actually single-ended.  At least in the case of the gear you described, the XLR output is in fact balanced.  It's just not done in the best possible way.  And it may sound fine, for all we know.  Furthermore, Denon tells customers what they are getting, if you take the time to read the not so fine print.
I refer back to the Sony SCD1; yes, it had that suboptimally derived balanced output, but it was also regarded for many years as the best sounding one box CDP available and is still sought after by some collectors and tweakers. 

Al, 1+

Denon appears to adjust their output voltages as they seem fit. It’s 500mV for both outputs of the DN-300H (otherwise, they would have stated so if they differed). It’s different for their internet tuner, but they do specify output level differences where they exist. See:
https://denonpro.com/products/view3/dn-350ui

I got the tuner today. I know why someone sold it. It hums. Not through the speakers, through its cabinet, primarily the top. A small amount of pressure at a single point on the top stops the humming. I can only hear it with the music muted and I have to be pretty close. From my listening position I cannot hear it at all even with the music off.

It does not seem like a malfunction, I think it is just poor design and construction. Probably not enough for me to send it back.

Otherwise it is like new and works great. 

@n80. Weird. 

Not so weird.  Probably only a transformer that needs either securing or tightening of the leaves.  Open up the cabinet.  If the transformer has bolts going through from one side to the other, tighten them.  Also, look to be sure that the transformer is tightly bolted to the chassis. Ideally it will be supported by some rubber-y footers.  Tightening everything up will likely rid you of the problem.  This thing is old, is it not?  Why are you so quick to dump on it, when in fact this is a typical problem that can apply to almost any vintage gear?  It may even be that someone tried to service it and left the transformer loose in the chassis.  If the two suggested cures do not work, you might be able to dampen the vibration by adding a weight on top of the transformer.