What does a subwoofer solve?

It is very unlikely that any hifi source will not provide enough base, bad base perhaps.  An amplifier may well be bass shy or lack bass control and certainly speakers are likely to have the biggest effect.  However a vital component affecting bass is the room. I have a small(ish) room with smaller floor standers and a 150 watt per channel hybrid power amp.  No source problem, no amp problem, no speaker problem but huge room problems. In the end stereo subwoofers and a minidsp have solved it, together with lots of room treatment.

Most likely issue is the room/speaker placement. Maybe no need for lots of purpose-made room treatment if you can move a few things around and there are not a lot of hard/shiny surfaces in the room. 

Keep in mind that we listen to our systems THROUGH the room.

Thanks again for all of the input.  I'll stand down on the subs for now and investigate some alternative placement of the room components, especially speakers.

One example of what I mean when I say they amp is shy on bass is Max Richter's Blue Notebooks album.  Second song (On the Nature of Daylight) just after the 1:30 mark on the song there is a sustained series of deep notes that you feel as much as hear.  With the BHKs, this was very present and audible.  On the Luxman, it's there, but barely.  This example translates to various other music.

I ditched the BHKs due to quality issues.  Three sets and they couldn't send me a pair that worked as advertised, so I gave up.

One more thing: The front end is a Linn Klimax DSM Katalyst (I know I'm in the land of mostly Linn haters here, but I can live with that).  Output impedance is listed at 300 Ω.

If you mean,’" What potential problems can a sub-woofer solve with regard to sound?" I would say many. And I would say in my own system, when properly integrated they supplement the low frequencies and allow me to hear lower frequency sounds. They add and enhance those lower frequencies because the sub-woofers I had chosen are flexible and designed to do just that. They can add lower level detail and allow my speakers to simply produce the frequency range they are designed to reproduce. I also think the allow my amplifier to work easier. in the end it’s a matter of your flavor of equipment and sound. Sub-woofers are just another way to get quality sound. But not always the best way. BTW even full range speaker systems have been known to benefit from quality sub-woofers. IMHO