Don’t get me wrong, PM is marvelous and there’s no gainsaying its enormous historical impact.
If your characterization of Metamath is correct, I don’t think that’s in the spirit of PM at all. One of the major problems PM had was the rejection of (what later became) the Axiom of Choice in favor of Russell’s convoluted theory of types. Indeed, that set theory (ZFC) is needed to get the rest of the way R&W were trying to go is one of the signal failures of the logicist program behind PM.
However, if you think the main point of Principia Mathematica was the very specific set of axioms that they chose, then that's different. The PM authors chose to use a "ramified" theory of types, which is complex. It does have sets, it's just not ZFC sets. Few like its complexities. Later on Quine found a simplification of their approach and explained it in "New Foundations for Mathematical Logic". There's a Metamath database for that "New Foundations" axiom set as well (it's not as popular, but it certainly exists): https://us.metamath.org/index.html
More Metamath databases are listed here, along with some other info: https://us.metamath.org/index.html
There are pros and cons for both systems.
The biggest problem NF has is, as usual, a social one: there just ain’t a lot of people working on, or interested in working on, NF compared to other set theories.
If you are interested in the underlying goal of Principia Mathematica, I urge you to check out the Metamath Proof Explorer (MPE): https://us.metamath.org/mpeuni/mmset.html
By itself, Metamath doesn't have built-in axioms. MPE uses Metamath to first state a small set of widely-accepted axioms, namely classical logic and ZFC set theory, and then proves a tremendous amount of things, building up to a lot of mathematics from careful formal proofs that rigorously prove every step.
Some things cannot be proven, but that doesn't mean that proof is dead.
https://us.metamath.org/mpeuni/mmset.html