(Here's the last word from Betty Azar about "reduced adverb clauses.")
Also, I think it's important to recognize that teaching modifying participial phrases as "reductions of clauses" is basically only a pedagogical means of demonstrating the meaning of such phrases. Rather than saying they are "reduced," one might more accurately say they are what you might call "equivalent structures" -- two structures with equivalent meanings. It's just that in teaching, one takes the known, in this case adverb clauses, and builds on that foundation to arrive at an understanding of the form, meaning and use of related structures -- it's what we do all the time when teaching grammar. In teaching grammar, one can take a pedagogical approach too far if one tries to turn it into a formula that will explain and account for every possible occurrence of certain structures in a language. But those of us who are grammar wonks LOVE to try to figure out how all this works!!