Ok, so here is your first issue. The movement of the distributor when you rev the motor is strictly a function of the vacuum advance. The mechanical advance has no effect on the movement. It is very important that both advance systems are working properly because each has its own separate but somewhat interrelated function.
The mechanical advance is underneath the point plate so it cannot be visually inspected without taking a few things apart. There are a couple of things that can prevent the mechanical advance weights from working properly. A lot of gunk, dirt and other crud that has built up over the years preventing the weights from moving, a buildup of rust, or the weights are rusted to the pins of the mechanism preventing them from moving, broken springs or broken pins, or a combination of the above.
To inspect, you will need to remove the cap and rotor, then the black plastic insulators, retaining stud and wires (point and coil). There are three screws holding the point plate to the distributor. Two of these screws also hold the spring clips for the cap. You remove the screws and clips, then the point plate. Now the advance mechanism can be inspected. Give it a good cleaning, and lubricate the weights and springs and pins. If anything looks amiss or missing or broken, time to have the unit serviced.
In the first photo, you can see where there is a advance weight spring missing. In this instance, the spring broke from the pin, and then jammed in between the weight preventing the weight from moving outward and thus preventing the advance from working. You have to look hard, but the pointer is indicating part of the spring that broke off and fell to the bottom of the distributor, the other part got itself jammed in the weight.
The second photo is what you are aiming for. Now, you will not be able to remove the shaft from the distributor without removing the gear. So for you, I would get in there with some carb spray and clean clean clean all the grease and junk. Then a little light oil (like sewing machine oi)l, and a little dab of grease here and there, and get the advance mechanism and weights lubricated, working or working better anyhow.
