Just realized that Yonatan Harpaz actually suggested this in a comment above, that I had forgotten about. Looks like we basically did the same thing!

Let us continue this discussion in chat.

@მამუკაჯიბლაძე I am inclined to believe that one cannot have a Hopf-algebra PROP, in the same way that one cannot have an operad for groups, but I might be wrong.

@მამუკაჯიბლაძე there are also, if I understand correctly, properads and $\infty$-properads for bialgebras, which might be slightly more general. I am just not aware of anyone doing this for $E_n$.

@მამუკაჯიბლაძე ah, I wasn't familiar with that, maybe I'll check it out.

@მამუკაჯიბლაძე compatibility conditions are NOT worked out anywhere. the best I can do is what I said above, but writing down what that means geometrically or combinatorially seems like an unbelievable mess.

I wanna know what happened to the person that wrote this...

In the case that $V$ is co-complete then the category of $V$-enriched symmetric multicategories (aka symmetric colored operads) is co-complete. This is discussed here: arxiv.org/abs/1111.4146, but it really is a straightforward generalization of the statements in the paper by Elmendorff and Mandell cited in that paper.

A silly comment, but X(1) is the sphere spectrum, so you probably don't want to include that in your list of examples of spectra that don't detect.

@DavidRoberts haha I guess I should, to be a good community member and all.