Why does the LHC use lead ions? The obvious answer is more energy per collision. This is discussed on p21 in this brochure of FAQs: CERN-Brochure-2017-002-Eng, which I found linked in this old question: Why not build a particle accelerator on ground level? What is the shallowest feasible depth to build one?.

Each proton beam flying around the LHC has a maximum design  energy of
7 TeV, so when two protons collide the collision energy is 14 TeV.
Lead ions have many protons, and together they give an even greater
energy: the lead-ion beams have a maximum collision energy of 1150
TeV. Both collision energies have never been reached before in a lab.

But from Accelerator Science: Proton vs. Electron, collisions are not between lead ions so much as two quarks/antiquarks/gluons within two protons or neutrons. Mostly the ions pass through each other.
Is it simply because more massive ions can be accelerated to higher energies per nucleon, as implied on p19?

In a circular accelerator, such as the LHC, heavy particles such as
protons (protons are about 2000 times more massive than electrons)
have a much lower energy loss per turn through synchrotron radiation
than light particles such as electrons. Therefore, in circular
accelerators, to obtain the highest-energy collisions it is more
effective to accelerate massive particles.

Or is it that you get more nucleons in the beam with ions?
Either way, why not use something heavier like uranium? And why are Future Circular Colliders at higher energies than the LHC planning to use protons?
 A: Maybe this article will clear up your questions.

Collisions of lead nuclei will allow physicists to study specific phenomena such as quark-gluon plasma, a state of matter that is thought to have existed during the very first moments of the Universe,

You ask:

Either way, why not use something heavier like uranium? And why are Future Circular Colliders at higher energies than the LHC planning to use protons?

The choice of colliding particles in an accelerator depends on the objectives of the study, which is to see the interactions of elementary particles  and test the mathematical model we have arrived to describe  them, also looking for deviations and new effects. The simplest the particles colliding the easier the test of the fundamental interactions. That is why e+ e- colliders were built, the results of those experiments validated and extended the standard model.

is it that you get more nucleons in the beam with ions?

for the study of quark gluon plasma the objective is to have as many nucleons involved in the creation  in order to reach if possible plasma energies. Lead is a stable nucleus . It would be foolhardy to use unanium or other heavy unstable nuclei, both for the mathematics of the analysis but also of the extra dangers of radioactive debris that results in all collider experiments.
Protons are the simplest carriers of quarks and the experiments can study a lot of interactions and decays of new particles with the weak force and also , from deviations to the theory search for new effects due to weak interactions. Heavy nuclei are for studying plasma as quoted above. I am sure that once a new collider exists, there will be experiments with heavy nuclei to study further the plasma.
There are also plans for future e+e- and even thoughts of gamma gamma colliders to study simply fundamental interactions.
It all depends on the physics objectives of the study.
