What does a subwoofer solve?

Brings more problems than benefits. Get a good full range and use the sub for HT only

a properly matched system will not make you aware of "missing bass" 90% of the time, but i have at least 3 pipe organ recordings that are fairly palpably loud below 20 cycles per second [locks the air in the room with a vigorous pulsation of air pressure] that NO standalone stereo speaker system i’ve heard can touch, not even Snell type As or KEF 105.2s with monster amplification. for those 3, a great sub is MANDATORY. i know of a retired doctor whose large house contains a massive digital pipe organ - to reproduce those 32’ contra bombarde pipes he uses 8] twin-12" acoustic suspension subs in massive cabinets. that amount of speaker enabled the visceral pulsating room-shaking sensation of real 32’ organ pipes to fully manifest.

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.