I do believe it is up to biochemists to demonstrate a plausible route to chemical evolution. The lack of such a route is why there is no theory of biogenesis.
I don't believe we will ever know the exact path to first life, even if we can demonstrate a plausible natural path.
There are several reasons why most, biologists believe in abiogenesis. One is historical and cultural. It was thought for years that organic compounds could not be synthesized, but eventually they were. It was thought for years that complex organics like amino acids could no arise undirected, but they did. It is reasonable to attempt further steps anong this line.
Another reason is that science has no choice. Science isn't in the business of proving things can't be reduced to natural causes. Just the reverse. Lack of success proves nothing.
I note that you have centered somewhat on the definition (or description) given with regard to "specified complexity". In my view not all definitions of complexity are equal. That particular one strikes me as vague.
If I were to choose, I'd select Kolmogorov for the least description and functional complexity for the least time. But all of them are helpful and give insight to the question at hand.
With regard to the biogenesis/abiogenesis question and specified complexity, the mathematician seems to be focusing on a need for an algorithm at inception - a source for information. I did not take his as a purely bio/chemical question.
In biology and I would imagine all sciences that deal with a historical record and incomplete information (archeology, anthropology, etc.) - the absence of evidence is not evidence of absence.
In mathematics and in physics, however, the absence of evidence is evidence of absence.
When the two disciplines look at the same question - e.g. abiogenesis - the one is not concerned about an absence of information and to the other, such absence would be dispositive.