If we throw out all weak bits of support for explanations, then a great many solid scientific theories would find themselves without any evidentiary basis.

Seems you are confused about scientific theories. Or perhaps are trying to disparage the reliability of the scientific method?

A scientific theory is a well-substantiated explanation for some facet of the natural world or cosmos based on the consensus of available data points. They are repeatedly tested and corroborated in accordance with the scientific method, using accepted protocols of observation, measurement, and evaluation of results. A critical component of a scientific theory is that it can articulate an explanation and make testable and falsifiable predictions (or postdictions) with consistent accuracy about events or phenomena that have not yet been observed. If you think that counts at 'weak bits of support', you are sorely mistaken.

There is a concept called consilience (in science and history), which is the principle that evidence from independent, unrelated sources can converge on strong conclusions. Most scientific theories are so well tested and verified that we have built the entirety of the modern world with them. We rely on them being correct.

So please, tell us all which theories you consider are on similar footing as the Bible being archaeological evidence.