Hello, I have the same problem with mine (HP 7510).
Symptom: the feeder goes down on paper. It makes some noise, but the wheels barely rotate and do not push paper through to the second set of wheels. If I put my hand under the feeder to stop it from going down and then push the paper manually, the second set of gears catches the sheet and the printer works fine.
After inspection, I found out the culprit: all the gears within the feeder are working fine, but the first one, attached to the rotating axis, must be to old and it is not stuck onto the axis anymore. The axis turns fine, but it does not get the gear to rotate as soon as there is a slight resistance (which is the case when it has to push paper).
I do not know how to repair that. I may try to glue the gear on the axis, but my experience with this is not very good as the rotation is "brutal": it starts and stops violently, so the glue can only last a short time.
Would anyone have an idea? Do you think we can get spare parts?
It is the second HP printer I have where a small 2c plastic parts breaks after 2-3 years, which eventually causes the whole printer to be trashed. Honestly, that is a shame.