A 2020 paper Operads, monoids, monads, and bar constructions by Ruozi Zhang, Foling Zou, and J.P. May revisits the foundations of operad theory and discusses several possible choices for what $\mathcal{C}(0)$ might be in an operad $\mathcal{C}$ defined in a symmetric monoidal category $\mathcal{M}$ with unit object $\mathcal{I}$. It might be an initial object (so empty in some examples, $0$ in others). It might be a terminal object (when the operad is often called reduced), or it might be the unit $\mathcal{I}$ (in which case the cited authors and others say that $\mathcal{C}$ is unital), so unital = reduced when $\mathcal{M}$ is cartesian monoidal, but they are quite different otherwise. So whatever May may have told his ex-student Shulman over a decade ago, he firmly disagrees now. David, much as you and May usually agree, this means that May also firmly disagrees with your sentence "An operad where $O(0)$ is the monoidal unit of $\mathcal{M}$ is called reduced.'' In examples where $\mathcal{M}$ is not cartesian monoidal, unital is much more accurately informative than reduced. The cited paper shows that unital operads admit an interesting variant of Kelley's interpretation of operads as monoids in a certain monoidal category, and that reinterpretation is seriously interesting in applications.
Saying that there is no $\mathcal{C}(0)$ says that $\mathcal{C}$ is not actually an operad, or so says the pigheaded person who first defined operads. The horrid if accurate term ``constantless operad'' should be viewed as a noun. That is, constantless should not be viewed as an adjectival modifier of operad.
