If you have any doubts about what your surgeon is telling you then you need to get a second opinion. If he/she feels the threads can and should be removed then that is your best guide. Having done a few of these procedures over the years, I can tell you the reason they are no longer available is that the lift only lasted a few months at best. Therefore, if should make little difference if they are removed four years later. With that said, I would not have a surgery simply to remove them if they aren't bothering you. If they are deeply embedded that could be more harmful than beneficial because of the trauma to the tissues getting them out.
Generally speaking what most people call a neck lift is really what we think of as a face lift. While there are a few less common operations out there that just treat the neck, the loose skin under the chin almost always accompanies jowling and should be they addressed together. Simple removal of loose neck skin is possible but even the best scar will be immediately apparent. Instead, the scar is hidden by the ear and loose neck and facial skin is shifted to around the ear where it can be removed with a more cosmetically acceptable scar. This would be a standard face lift. Combining it with eyelid and brow rejuvenation and you have what most people refer to as a full face lift.