Is it true that reducing feedback factor or gain-bandwidth product could reduce power consumption ?
Gain-bandwidth product of an amplifier by itself - yes, you can lower power consumption and it will probably lower the gain bandwidth.
Don't see what feedback factor has to do with lowering power consumption. The lower the feedback factor, the higher the gain, so for a given input, the higher the output voltage amplitude. That would increase consumption a little.
Slowing down an amplifier can mean you can have much higher DC gain in the amplifier if you want it stable at all gains. If you take an amplifier that has a pole at 10 Hz and another at 1MHz, you probably want to have the 0dB point below 1MHz so you keep away from the chance of instability, so that means you do not want the amplifier gain to be more then 100,000 (100dB). If you can move the first pole to 0.01Hz while leaving the second pole at 1MHz, you can now have a gain of 100,000,000 (160dB) and be stable. This is great for precision circuits where you do not need speed. The cheapest way to reduce that first pole frequency is to lower currents in the first stage. This probably also reduces input currents. The other way to slow the amplifier down is to make the miller capacitance in the second stage 1000 times bigger and that is very expensive in an IC design.