I'll just clarify some points the came up within your questions.
"Yes, pushing off actively works because the ground is solid and will provide a nearly equally strong force into the opposite direction, i.e. forward and upward."
Not nearly equal, EXACTLY equal.
"But why shouldn't it be possible to apply a force which acts in this direction by pulling the foot actively from the ground?"
Yeah, this bit is hard to follow. A force must be applied, that should be clear. You just can't apply a force to the gound by pulling your foot away from it - just Newton's laws again.
"Provided that the leg is bent while being in contact with the ground, a contraction of the hamstring muscle will further decrease the angle between thigh and calf."
Fine.
"If the force is strong enough you will break the ground contact"
if the raising of the feet with respect to the COM is quicker than the falling of the COM.
" and if you at least maintain the vertical position of your body while the leg(s) are moving slightly upwards then your general center of mass will move up."
Here's the problem. If all you did was pull, then you would not maintain vertical position of your body. When you feel that you are doing this, you are not. Either you are dropping COM height without realising it, or you are giving a push prior to the pull without realising it.
Hope this helps to clarify