size of the driver

Coming from the "old school" and being a complete (or almost) novice here - in the old days back in 1970-75 we thought that the big (read - wide) driver will have better capability to produce more realistic sound, talking about lower end of the spectr at least. But nowdays I am seing 6 inch drivers stated as "bass". Just curious how well those perform or in another words what is the secret behind those if they really can perform at the same level as the 12 inch ones?

And another question which I guess is too simple and too basic around here that's why I couldn't find some point to point answer - when we speak about sensitivity - would that be a true statement to say that higher level (say 92-95 db) will allow to extract "fuller" sound spectr at a lower level of volume? My feeling is that with somewhat lower 89 db or less you need to increase the volume in order to have more visible lower end?

Thank you for your time