Rather than the experimental approach to accreting "proof" incrementally, it may be more fruitful to take the Aristotelian approach, and simply assume a Prime Mover or First Cause of everything that is, and then see if there's anything we come across that disconfirms or refutes our universal premise.
That would make sense to me, too; however as you say But this would be the very approach that is most strenuously avoided these days as thoroughly "unscientific."
On the very long thread I offered a hypothesis with methods of falsification, as follows:
Falsifications: That such algorithms or information content do not exist - or that such algorithms or information content can arise from null.
After about 3000 posts on the big thread (and a lot of research since) - I remain convinced it is a good layperson's scientific hypothesis to determine intelligent design.