Your cruising speed at a given HR will increase. And that happens fairly fast. If you think that is different from "fitness" then I'm tempted to ask what you think fitness actually is.

However, don't get too hung up on the numbers. Lots of power-based trainers will boast to you about their FTP but can't race their way out of a paper bag, because success at racing is down to a lot more than the numbers. In fact I'd go so far as to say that the strongest rider loses as often as not. Lots of strong guys can manage fast but one-paced TT type efforts but can't cope with the repeated accelerations that a bike race typically consists of, or lack the brains to spend enough time out of the wind, or waste their strength trying to chase down repeated breaks that aren't going anywhere, or whatever.

As far as progress is concerned, believe it or not there was a time before HR monitors. Eddy Merckx didn't have one and he managed pretty well. They trained on the basis of time on the bike, and gear selection, and they measured their progress with a stopwatch - timing themselves over a prescribed course. Merckx himself spent many many hours being paced by his coach on a motorbike - three hours at 25 mph, that sort of thing. Their most important measures of progress, of course, were race results.

With a HR monitor it's pretty straightforward to create a training plan. Test for your LTHR then create HR zones based on either Friel's or Coggan's system. Then train to thise zones as you would to power zones. Friel's training bible explains how to adapt his plans for HR training and hiw to assign TSS to HR-based sessions. HR has limitations at the top end because of lag, and (to a lesser extent) on long sessions because of cardiac drift, but these can be compensated for with experience.