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.

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.

Edit in response to request from the OP:

The conditions of the theorem state that $M$ is locally presentable, and

1. $W$ is $\kappa$-accessible for some $\kappa$,
2. $W$ is closed under retracts,
3. morphisms in inj$(I)$ are weak equivalences,
4. within cof$(I)\cap W$, morphisms with cofibrant domain are closed under pushouts of diagrams of cofibrant objects, and are closed under transfinite compositions,
5. The maps of $I$ have cofibrant domain and the initial object of $M$ is cofibrant.

Then you have a combinatorial semi-model structure with generating cofibrations $I$, generating trivial cofibrations $J$ constructed as in Barwick or Beke's papers, cofibrations cof$(I)$, and fibrations rlp$(J)$.

I should be done with this paper by the end of 2019. Right now I'm just adding lots of examples.