It sounds like it could use some pruning to remove the dead growth. If water is pooling when it rains, then there is a drainage problem unless you got a lot of rain all at once. It probably isn't happy if it is sitting in water for long periods of time; on the other hand, it may be being kept too dry if it is only getting water when it rains. I noticed that my M. figo wants to be moist all the time, but I have it growing in a container. If I let it get too dry, it will drop its leaves and have some dieback. It also sounds like it needs some fertilizer for acid-loving plants like Miracid and some Iron chelate for the leaves. You could move it, but M. figo's resent being transplanted, and it may not survive the transplant.

A lot of the limbs look practically dead, but have leaf buds and flexibility. I can go out every few weeks and just flick the dead parts off.

Water will come off the patio, flow around the base of the plant and then drain off down the edge of the flowerbed and out of the backyard. Because of the drainage, the soil around the plant is hard on the surface. I'm not sure how much water is even getting through that top layer.
(So moisture is one thing that needs to be worked on.)

Would fertilizer for azaleas and gardenias work?
Should it be treated more like it is an azalea?

I'm wondering if it could be getting too much sun, or if the patio heat is a problem.

When I read that it is a shrub and can be used like a ligustrum, I couldn't believe it. It simply doesn't have enough leaves.

It does sound like it is not getting enough water. It should receive a good soaking if water is not getting below the surface. You may have to surround it with rocks or dirt or do something to keep the water from running off. The fertilizer for azaleas and gardenias is fertilizer for acid-loving plants and would work fine. Heat shouldn't be a problem, but M. figo's may prefer filtered sun to full sun, but I still wouldn't transplant it for fear of losing it. If it has been in the same place for ten years, it can survive there and probably just needs more water and more attention.

I think I'm having similar issues with my figo. I bought it at a reputable nursery this spring. When it was purchased it was about 3' tall and 1 1/2' round. It was full of leaves and blooms! It looked like a small bush. My wife planted it, between two small river birches that we also had purchased, in a new bed. She used cow manure and potting soil in hole with plant. Our ground is slightly acidic and very clayful. We get abundant rain and very high humidity. My birches are doing fine but my figo looks horrible! Not long after planting it, we lost all the blooms and most of the leaves. This is the most leaves it has had. They sprout green, get full size, turn yellow, and fall off. The branches are brittle and the ones that don't have leaves seem dead. What should I do to save it?

Michelia figo do not like to sit in water they have fleshy roots that will start to rot which is not normally a problem outside. They like a free drainage soil/compost and a good soaking they also love the shade and hate the hot full sun which will tend to bleach the leaves and turn them a anaemic yellow they do best with early morning or evening sun.
I would recommend feeding DYNA-GRO Foliage -pro at half strength every other watering ( little and often is better then a large amount in one go ) you can use it as a foliage spray or in the soil. Foliage spray is prob best easier and quicker for the plant to take in and is best done at the evening.
You could also try then on Banana peels cut some up and leave them in a bucket of water for 48 hours then use the water to feed the plant.
But after saying all that it might not be your fault they can suffer from a nematodes that attack the roots.
Hope this helps in some way and you can save your figos because the scent alone is worth trying to safe them i only wish i could grow mine outside here in the UK but they all have to be grown in pots :(