How much is concentration a part of our practice while sitting?

All the Buddhist stuff I've read over the years seems to clearly imply, if not come right out and say so directly, that sitting is supposed to increase your concentration over time. That is, while sitting, you will eventually be able to concentrate better and for longer and longer periods of time, to focus more on and reside in that still space. Well, I've been sitting for four years and it seems my mind wanders as much as it ever has. I feel no increase in concentration (singular focus) while sitting at all. I feel I have gained much from my practice, but an increase in concentration level or quality does not seem to be part of those gains. This has begun to bug me, thus disturbing my practice. Today while sitting I realized that this was a desire that was causing suffering, so I let it go. Ahh, OK. Then, because I couldn't concentrate, I realized that I needed to let go of the idea of concentration entirely. Ahh, this was very good! But then, because I couldn't concentrate, the commentary came back to say, "no, really, what about concentration?"

Am I missing something? Should I be doing something different?
Is this even an issue? Shikantaza being letting thoughts come and go seems to imply that there is no concentration, sort of thinking non-thinking, right? So is it concentrating on not concentrating? But then have I misunderstood all those references to concentration?
Is concentration even the right word? Is awareness a better term? Because that has increased a great deal over the years. I think I can say that I am able to concentrate on being more deeply aware, but my mind still often wanders away from awareness.

OK, I better stop here. So, what's the deal?