How much of the proton's mass is due to the Higgs field?
My view on this is coloured by Gian Francesco Giudice’s 2010 book A Zeptospace Odyssey. On page 173 he said this: “The most inappropriate name ever given to the Higgs boson is ‘The God particle’. The name gives the impression that the Higgs boson is the central particle of the Standard Model, governing its structure. But this is very far from the truth”. He also said “the Higgs sector is rather arbitrary, and its form is not dictated by any deep fundamental principle. For this reason its structure looks frightfully ad hoc”. Giudice also said the mass of a body is the intrinsic energy of a body at rest, and tells us about E=mc². In his 1905 paper does the inertia of a body depend upon its energy-content?, Einstein said “the mass of a body is a measure of its energy-content”. No problem there.
The problem is that the Higgs mechanism contradicts E=mc². Is the mass of a body a measure of its energy-content, or a measure of its interaction with some space-filling field? The idea for the latter comes from the electroweak sector of the standard model, where the weak force is said to be mediated by massive vector bosons. They have to be massive because the force is short range. See the Wikipedia Higgs mechanism article where you can read that “according to Goldstone’s theorem, these bosons should be massless”. If they aren’t, the bosons aren’t gauge bosons, and the theory isn’t a gauge theory. But here we are. We have Giudice saying “the Higgs mechanism accounts for about 1 per cent of the mass of ordinary matter, and for only 0.2 per cent of the mass of the universe. This is not nearly enough to justify the claim of explaining the origin of mass”. He also said the Higgs boson is “like the toilet of the Standard Model edifice”. As you are doubtless aware, Giudice is now head of theory at CERN.
The proton mass is 938 MeV. People often claim that (A) The proton is a bound state of two up quarks and one down quark, with the three quarks contributing a total rest mass of $2 \times (2.2 \text{ MeV}) + (4.7 \text{ MeV}) = 9.1 \text{ MeV}$ (about 1% of the proton mass), and the other 99% coming from "QCD binding energy".
I see you've read Matt Strassler's article what's a proton anyway? The above claim is the original quark picture. That has morphed over the years. This picture from physorg hints at it:

Image from the PhysOrg article Physicists zoom in on gluons' contribution to proton spin
People further claim that (B) The "QCD binding energy"'s contribution to the proton mass is independent of the Higgs field. Therefore, only 1% of the proton mass comes from the Higgs field.
Yes, people like Gian Giudice.
Claim (A) is perhaps defensible as a heuristic explanation, although it sweeps a lot of subtleties under the rug. (For example, in a non-QCD context, binding energy (as usually defined) always decreases a bound state's energy below that of its constituent parts - the opposite of what happens in hadrons.) But as I understand it, claim (B) is simply incorrect.
I think claim (A) is wrong, and I think claim (B) is wrong too.
As explained here, the proton ground state is best thought of not as a bound state of exactly three quarks, but instead as a superposition of many different huge collections of quarks and antiquarks, each of which has total up quark number 2 (i.e. two more up quarks than up antiquarks) and total down quark number 1 (i.e. one more down quark than down antiquark).
Matt Strassler said the proton isn’t made up of three quarks joined together by three gluons. He says that’s a lie, a white lie, but a big one. Instead he said there’s “zillions of gluons, antiquarks, and quarks in a proton”, and gave a picture of a whole host of quarks and gluons, all mixed up together like beans in a bag. All “rushing around as fast as possible, at nearly the speed of light”. But in low-energy proton-antiproton annihilation, do we see all these quarks and gluons come spilling out? No. What we see, is light:
Image credit CSIRO, see The Big Bang & the Standard Model of the Universe
This is sometimes described as "two 'valence' up quarks and one 'valence' down quark, surrounded by a huge 'sea' of quark-antiquark pairs". In my opinion, this description is also misleading, because it implies that there are three specific "valence" quarks that are "real", while the other "sea" quarks are just "virtual" and physically distinct from the valence quarks. In fact, every individual quark is physically on the exact same footing. In particular, I believe that all of the quarks gain a mass contribution from the Higgs mechanism.
I share your sentiment that this description is misleading. But I will also say this: if all those zillions of quarks are physically on the exact same footing, why have you never ever seen a single one? What else do you think might be "rushing around" at the speed of light?
As Prof. Strassler explains in the link above, the proton mass is best thought of as arising from the sum of three contributing terms:
1. The sum of the rest energies of all of the (many!) quarks and antiquarks
2. The kinetic energy of the quarks, antiquarks, and gluons
3. The binding energy stored in the gluon fields, which is negative and actually decreases the proton mass relative to what it would be if the quarks did not interact.
Note this in the Wikipedia gluon article: "as opposed to virtual ones found in ordinary hadrons". The gluons in a proton are virtual. As in not real. Also note that it's the wave nature of matter, so check out what Martin van der Mark said in on the nature of stuff and the hierarchy of the forces: smaller mass means bigger wavelength, so you can’t fit a longer-wavelength 2.2 MeV quark inside a smaller-wavelength 938.27 MeV proton. Last but not least, remember what Einstein said: the mass of a body is a measure of its energy-content.
I believe that the Higgs mechanism is responsible for the contribution (1), which is much larger than the naive 9.1 MeV from just three quarks, and therefore much more than 1% of the proton mass. Is this correct?
I think not.
Also, do we have any kind of quantitative estimate of the contribution of each of these three terms? The number of up quarks and the number of up antiquarks are individually indefinite, but can we numerically estimate their ground-state expectation values in order to estimate (1)? I know that QCD is strongly enough interacting that it's not particularly useful to think in terms of individual quarks, but I'm curious whether anyone's attempted that calculation.
Sorry, I can't answer this part of your question. But I share your view that the 1% figure is wrong. Only I don't think it's too low. I think it's too high.