How acceptable are the outputs of automatic LaTeX converters for journals with LaTeX requirement?
The most probable answer to what acceptable is the conversion from word to LaTeX is probable too little. The main reason is that word is more based in local format (visual appeareance) rather than semantic tags. 

For instance, you need to specify that a group of specific paragraphs are the abstract. LaTeX made this using an instruction like 
\abstract{ ... text ...}. A converter needs to look in styles to learn the meaning of the text fragment against its list of heading, leyends, abstract, section, subsection, etc.

But if what you want is have a simpler medium to wrote and rewrote contributions to the text, provided that you have a a solution to create graphics in jpg, png or pdf format and a transfer the more complex tables to LaTeX way, you can use Pandoc to convert from word to LaTeX.