In 1997, Elmendorf, Kriz, Mandell, and May wrote a book *Rings, Modules, and Algebras in Stable Homotopy Theory* in which they introduced the category of S-modules as a model for the stable homotopy category. The category of S-modules is a closed symmetric monoidal model category whose monoidal product descends to the usual product on the stable homotopy category. Its unit is the sphere spectrum. All objects are fibrant but the unit is not cofibrant. Every operad O is admissible, meaning the category of O-algebras has a transferred model structure where a morphism $f$ of O-algebras is a weak equivalence (resp. fibration) if and only if $U(f)$ is in S-modules. A reference is Proposition 1.5 in "Moduli Spaces of Commutative Ring Spectra" by Goerss and Hopkins. The proof uses that S-modules have a structured interval object.

For the operad $O = Ass$, Theorem VII.6.2 in EKMM proves that if $A$ is a cofibrant S-algebra then the unit $S \to A$ is a cofibration of $S$-modules. Just after, they write "In the commutative case, the argument fails because we must pass to orbits over actions of symmetric groups."

In modern terminology, this result proves that the operad $O = Ass$ is *strongly admissible*, a notion [studied by Pavlov and Scholbach][1].

The property that I want, that a cofibrant O-algebra should forget to a cofibrant spectrum, has sometimes been called "convenient" but that word is overloaded. Unfortunately, to get from "O is strongly admissible" to "cofibrant O-algebras forget to cofibrant underlying objects" Pavlov and Scholbach must assume the unit of the base model category $M$ is cofibrant, so this doesn't help with S-modules. I don't know whether S-modules satisfy the Pavlov-Scholbach condition of "symmetric h-monoidality" or the related conditions in Section 6 of [Bousfield Localization and Algebras over Colored Operads][2]. If they do, then the latter reference proves that cofibrant O-algebras forget to cofibrant S-modules for operads O whose spaces O(n) are cofibrant.

There are several other good models of spectra, including symmetric and orthogonal spectra, and various tweaks to their stable model structures. In the positive model structure on symmetric or orthogonal spectra, all operads are admissible. For symmetric spectra, this is proven in Theorems 8.3.1 and 6.1.1 of [Bousfield Localization and Algebras over Colored Operads][2], among other places. For orthogonal spectra this is proven in Corollary 5.15 of [Right Bousfield Localization and Eilenberg-Moore Categories][3], among other places. However, in general, a cofibrant commutative monoid need not forget to a cofibrant object in these categories, by Proposition 4.2 in Shipley's paper [A convenient model category for commutative ring spectra][4]. For this reason, Shipley introduced what is now called the *positive flat stable model structure* on symmetric spectra (and the same works for orthogonal spectra, see Remark 5.14 in the Right Bousfield paper above) where, indeed, cofibrant algebras over entrywise cofibrant colored operads (and commutative monoids) forget to cofibrant objects in these positive flat model structures, e.g., by Section 6 of the Left Bousfield paper above.

> (1) Can we tweak the EKMM model structure on S-modules in some way so that cofibrant commutative monoids forget to cofibrant S-modules?

It is worth remarking that one really does need to tweak the EKMM model structure to get a positive answer to (1), because S is a cofibrant commutative monoid that is not cofibrant as an S-module.

> (2) Can we do the same for some class of operads? Offhand, I don't even know that it's true for the class of $\Sigma$-cofibrant operads.

[1]: https://arxiv.org/abs/1410.5675
[2]: https://arxiv.org/abs/1503.06720
[3]: https://arxiv.org/abs/1609.03635
[4]: http://homepages.math.uic.edu/~bshipley/com4.pdf