You like the speed but worry the output carries the fingerprints of machine writing.
Omni's house style avoids em dashes, one of the clearest tells of AI-written text, so output reads more human by default. The rest is a matter of register and model: the default register of most models is stiffer than a person writing quickly, which is exactly what the tone actions exist to fix.
Run drafts through Direct or Casual before publishing, and point the text tier at a model whose voice you like; the model used for text actions is configurable in Settings, Models.