Okay, re. aero bars.
Balance/Control/Handling: Whilst you should not have excessive weight on the aero bars, there will inevitably a forward shift of weight over the front end, this will negatively impact bike handling and control, leading to instability. TT bikes get around this by having a slacker head tube angle and/or more fork rake, so that the weight is still far enough behind the front wheel hub in order to handle in a stable way, particularly in a straight line and through moderate bends. The narrow grip, forward shift of weight and no access to the brakes makes cornering on the aero bars a challenge, for most corners you would get OFF of the aero bars and onto the base bar/bull horn for control and access to the brakes. When climbing, if you need to get out of the saddle, you also need to get off of the aero bars, reason, the bike will rock side to side, to output most power and maintain stability, you must be able to resist the rocking with your upper body, this requires a wider grip. Only on low gradient climbs, such as long drags of something like 2-5% would you typically ride on the aero bars, because you are able to remain seated for the duration of the climb and would be going fast enough to get a benefit from the aerodynamic position.
Power on aero bars: Riding on aero bars would in most cases will cause the hip angle to become more acute, limiting power output, not increasing it. BUT, the aerodynamic benefit often outweighs the drop in power output, provided you do not close the hip angle too much. Again, referring back to bikes made to be fitted with aero bars, they have a steeper seat tube angle, shorter top tube and shorter head tube in order to rotate the rider forward around the ankle joint, getting them low at the front without closing the hip angle. You will also notice TT riders pull themselves forward on the saddle, this opens up the hip angle. Generally, you are aiming to preserve as much capacity to output power whilst reducing frontal area, it is a compromise and in most cases some loss of power will be inevitable (unless you are a specialist who rides in this position most of the time thus being most well trained to operate within this hip angle envelope), but so long as the aerodynamic benefit outweighs it, it is game on!
Re. Training Metrics
Power is the ultimate metric because it actually tells you how hard you are working. It doesn't matter whether you are riding into a head wind, up hill, down hill, on the flat, it doesn't matter if it is hot or cold. If you are putting out 200W, you are putting out 200W, whether you are going 20mph on the flat or 12mph up a hill, you are doing the same amount of work. This makes it both the ultimate pacing tool and also the ultimate training tool (both in real time and in post-ride analysis, you can determine a remarkable amount of information from this metric). Speed is a piss poor metric to measure on most turbo trainers because the resistance curve varies day to day due to various reasons, such as differing temperatures, so your speed will vary day to day for the same power output. If you go faster one day, vs the previous day, you have very little idea if you have gained fitness, or if the turbo trainer resistance curve has changed due to it being warmer or colder.