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Yes, and this is one of the main results in a paper I hope to put on arxiv very soon. I wrote about this result in a previous mathoverflow answer here. You are right that the way to do it is to focus on maps between cofibrant objects. This result was also known to Clark Barwick, and probably to Cisinski, as I discuss in my other answer. That's one of the reasons it took me so long to write it up. I didn't know why anyone would care. But, in the paper I'm finishing, there are tons of applications of this result. If you want to correspond more, I would be happy to.