Long-term Mathematica user (physicist) here:
I don't think the use of open source software would make most science -- maybe that does not apply to cs/datascience -- more replicable. Usually that takes an expert in the field. And usually these experts are employed at universities where Mathematica licenses are not the prime cost factor. That said, I am all for open source software. Although I would argue that probably trustable scientific results do not rely on the inner workings of e.g. Mathematica anyway, but use Mathematica as vehicle for say linear algebra or symbolic manipulation etc. While the inner workings of Mathematica may not be open source, in principle the relevant algorithms are not propriertary but usually well-known mathematical results and as such at least in principle easily reproducible outside of the ecosystem.
>> ...subject to our assumed framework and chosen prior.
This is usually implicit in such analyses.
>> More generally, what's even the point of placing a confidence level on a non-falsifiable claim? What utility does it serve?
It's not about falsifiability in the sense that someone should go out and measure the predicted lifetime of the universe experimentally.
The point of such lifetime calculations (and their associated confidence levels) is more as a test of the underlying theory - the Standard Model of particle physics. If this theory would make the prediction that the lifetime of the universe should be shorter than the observed lifetime of the universe, then clearly there is something wrong with the theory, or with the way the lifetime has been computed, since clearly we are still here to make that calcuation.
Usually, this is seen as a sign that there should be some new physics - i.e. physics that is not described by the standard model in its present form - that should enter at an energy scale relevant to the calculation. Since this we do not know about this physics yet, the calculation is necessarily incomplete, which would explain finding a "too short" lifetime of the universe, while with a more complete theory one would (hopefully) find a result, that explains observed lifetime of the universe.
The fact that the paper finds a "long enough" lifetime of the universe, could be seen as an indication that the theory is able to make predictions at energy scales relevant to this calculation.
Why the hate for PDFs, I can see why you would not want to communicate dynamic content with them, sure. But for academic papers, which are usually static text and equations and maybe some plots, PDFs do exactly what you want: At least in principle they look the same on every system.
If I want to email them to somebody, store a copy, read them offline on a reader etc. I have to deal with one single file, in a format that just works on many readers. What am I gonna do when I typeset a paper with webtechnology ... fire up a local server to display some inline math in my html page .. no thank you.
I agree that maybe layouting based on physical paper is maybe not ultimately necessary, but it gives the reader a familiar structure. The way the advertised web site is transferring the papers into a long scrolling list of text ... I find it rather disorienting and unstructured. Text that is split up into "pages" (whatever size they are in the end) somehow helps break up the reading flow.
In the end it remains to be shown that the gain from having academic papers not typeset in PDF outweighs the hassle of having to deal with non-standardized ways of rendering properly formatted text on websites (thinks like MathJax etc. do not support everything that is available in full LaTeX etc.).