Why is the mass of the proton such a precise value? Why is the mass of the proton such a precise value?
A proton is composed of 3 net valence quarks and what is often described as "binding energy" or "a zillion gluons and quarks and anti-quarks self annihilating and popping into existence". The quarks are only about 1% of the mass but the mystery to me is understanding why all of this 99% "amorphous" dynamic binding energy in a proton (or neutron) amounts to the consistent and very exact mass or energy that it is.
I would have maybe guessed that an individual proton's mass would just have more of a range, that the amount of binding energy would be more variable
 A: The mass of the proton has been measured to be $938.27208816(29) MeV/c^2$, the value in parenthesis the error in the  measurement.
The job of a theoretical model in modeling the proton is to attempt to explain the measurement.

what is often described as "binding energy"

It is not a good description, as  if you imagine an atomic type model just more complicated because of more particles. The strong interaction is involved in modeling hadrons, and the simple solutions and even the tool of quantum field theory is useless due to the large coupling constant of strong interactions.
The theory of QCD on the  lattice

Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum QCD is recovered

It  has managed to model the hadronic spectra given some inputs. Here is a presentation of the status of the theoretical model.
The hadronic spectrum  is research in progress both experimentally and theoretically.

To find out the origin of the proton mass experimentally is one the primary goals of the upcoming electron-ion collider. This work has opened the door for more numerical calculations and theoretical understanding of nucleon structure, including the spin decomposition of the proton in terms of quark spin, quark orbital angular and glue angular momentum, whose measurement is also a leading goal of the electron-ion collider.

A: The origin of proton mass is rather a subtle issue, it is really created by a complicated dynamics of quark and gluon fields. As far I understand this issue, the known value for the proton mass is an energy of a proton ground state - which is some eigenstate of corresponding Hamiltonian, excitation in the QCD theory. There are excited states with higher energy, mass correspondingly. However, in case of not very high temperature and density - the ground state will strongly dominate.
A: In quantum mechanics, and energy eigenstate is a stationary state: that is, unchanging with time (other than a phase). This picture doesn't work with the idea of virtual particles popping in and out of existence, and that is a problem with overly-classical analogies.
A: You say:

a zillion gluons and quarks and anti-quarks self annihilating and popping into existence

and while this is a very common way to describe the interior of a hadron like a proton it is actually rather misleading. Nothing is popping into existence then disappearing again. But explaining what is actually happening is a little involved.
Our current best theory for describing particle is quantum field theory. In this theory the fundamental objects are quantum fields that exist everywhere in the universe. Particles like quarks are not fundamental objects. Instead they are just states of the quantum field. This nicely explains how particles can be created and annihilated at colliders like the LHC, because we can start with a zero particle state of the quantum field and add energy to it to excite it to states that correspond to non-zero numbers of particles. Likewise a state of the field that corresponds to particles can decay to a state with fewer or no particles.
But while there are states of the field that do correspond to what we call particles, this is actually a rather special case. Specifically this is only the case when we have an isolated particle that isn't interacting with any other particles. These are called the Fock states of the field. But the field has an infinite number of other states that aren't Fock states so they don't correspond to particles. The problem is that we don't know how to solve the equations of the field to get these states. Instead we have to use approximate methods to calculate properties like their mass.
And this is the case for the bound states we call hadrons. A proton is a state of the quantum field but it isn't a Fock state. In principle we could write down the equation for the field and solve it to get the state corresponding to a proton, but in practice we simply don't know how to do this so we have to approximate it. We do this by approximating the state as a collection of virtual particles, and this is why popular science descriptions talk about particles popping into existence and disappearing again. Where the popular science articles go wrong is that these virtual particles are a computational device and they do not exist. I cannot emphasise this enough: the virtual particles are just a way of calculating the properties of field states that are not Fock states and therefore do not correspond to particles.
This has taken us a long way from your question, but we can now understand why the mass of a proton is well defined. It is because it is a well defined state of quantum fields and as such has a well defined mass. It just doesn't correspond to a well defined number of particles, which is why it isn't just three quarks or $n$ quarks and $m$ gluons or any other collection of particles.
If you're interested in finding out more about this you might want to look at my answer to Are vacuum fluctuations really happening all the time? where I use a similar argument to explain why the vacuum isn't actually fluctuating either.
A: A particle, be definition, is an energy eigenstate. That means that the state is constant in time.
The "vacuum", i.e. the state with no gluons and quarks, is not an energy eigenstate when interactions are included, because, as you say, quark and gluons will pop in and out of existence. But a bound state of quarks and gluons is an energy eigenstate. That means that there is a way to combine quarks and gluons in such a way that when the interactions are turned on, the stuff annihilating is exactly replaced by what is being created and the state is constant. One of those states is called a proton. And the energy of that state that stays constant is called the proton mass.
