I am afraid most information you find through Google will be very simplicstic... Real life is a bit more complex, probably even too complex to discuss it in a forum with the restricted possibility to draw up some graphs.
profile drag should only increase with speed
Unfortunately if we talk transport airplane we can not ignore mach number. In typical cruise the upper wing surface is 30-50% supersonic and profile drag is strongly influenced by this. A little change in lift may change transsonic drag significantly.
If we strictly talk subsonic, turbulent and moderate Cl your simplification is correct. If we go transsonic, consider laminar airflow and go closer to Cl max it becomes quite complex.
As Cl is limited, a weight increase typically means a speed increase (starting already before you even fly with Vr calculation...), this makes the induced lift discussion a bit academic. Nobody would (only) increase Cl to compensate for heigher weight.