Increase in power

My point is that coming home from a ride and saying "I went 1mph faster than last time = I must be fitter" is pointless. Wind speed and direction, the wheels you use, your clothing, ambient temperature, there are a dozen factors that affect speed that have nothing to do with your fitness.

In a road race, why don't you see if you can hold the highest speed for the whole thing and see if you come first....

Yes of course speed matters if you're a time triallist in a race. Fastest avg speed wins. In RRing, not so much. Work at getting stronger and faster together, but don't get hung up on training speeds as a measure of improvement.

Fwiw I do better at road races where the avg speed is lower. Lower speed tends to mean harder courses, more climbs, smaller bunches. Any race where the bunch averages 25mph+ is one where I'll be mid-pack at best as that means bunch sprint. So speed matters to me in that I seek out tougher races with slower average speeds. But I certainly don't treat training speed with any importance.

If you're in a road race and you finish with the highest average speed then of course you win. :p

Speed in context is what I'm talking about.

If you're in a bunch gallop, the faster sprinter most often wins and such speed can save them from the odd tactical faux-pas. Example: Cav and Greipel. It's believed the German puts out mammoth power, apparently a few hundred watts more than Cav but Cav has more wins because he typically covers the last 250m faster.

Need to kill it on the hills? I'd take a guess and say that Cancellera puts out more power than Quintana but who nails gets up those 8% grades first?

Speed in training can be used, if you look at trends, as an indicator of progression. It may not be all fitness - you may have made other changes such as weight, aero, position that alters long term comfort and ability to put out power over a long duration better etc etc. Even simple things like Strava KOM's on hills - you start heading up those leader boards then you're progressing. Knowing why you're going faster helps, but just like in races, it's the actual going faster up that hill that's the important end result, not that you're hitting a particular number (watts). The reverse is true for the ultra endurance guys - they could ask "I know I can put out this power for this amount of time, what can I do to go faster?" Again, speed not power.

Power is a fantastic metric for training and learning how to race, especially if you can't figure out when to and when not to go hard but there'll be times when you're hitting your max numbers for given duration during a race and it ain't enough because you're too slow - at that point it's just another metric to analyse to help figure out why you weren't fast enough. You need to pair power with weight and aerodynamics to have any true meaning.