Why is it then that if the BW restriction is active and significantly higher frequency sine than the limit is connected (let's say 2x the BW) the display cannot be filled vertically with the signal?
I'm not sure if I understand the question. If the bandwidth limit is set to 350MHz and you connect a 700MHz signal, there will still be a sine, but its amplitude will be much lower than expected (way beyond the 3dB point).
Whether you can fill the display or not is a matter of scope settings, but still the amplitude (absolute voltage) of the signal will be too low.
So I've assumed that a pure filtering loss in the analogue side can be compensated with increased gain.
You overlook the frequency dependency. The higher the frequency the smaller the amplitude. Since every signal is composed of a superposition of sinusoidal signals of different frequencies (Fourier series), a bandwidth limitation will remove the amplitude of higher frequency components. I.e. it will remove sharp edges from signals etc.
Another issue is that based on datasheet the afformentioned chip's -3dB point is at 900MHz. A bit low for 1GHz and a bit on the high side for 500MHz.
So what? The variable gain amplifier used in the SDS2000X series had a limit of 750MHz and the scopes were limited to 300MHz. Obviously you don't want the bandwidth of the amplifier to be a major factor in the system bandwidth, so choosing something with roughly twice the bandwidth seems somewhat reasonable.