1:28 PM
@User1236262625 I think it should be possible, but it becomes a "scheduling" problem -- you've got to make sure you eventually add in all the lifts which make their way in there in the standard argument, and they all have to be there at once when you hit each sufficiently-large regular cardinal... In theory there's no reason to do this, though, because if your cofibrations are closed under cobase-change and transfinite compositition, then they are closed under coproducts.
The kind of issues you need to deal with appear when constructing Fraisse limits.
@SaalHardali Somebody asking Emily and Dom about this at Talbot last year, and what became clear is that there should be multiple ways to do it. Dom's favored approach is the "operads as analytic monads" approach. I don't recall Emily having so strong a preference, but she pointed out that the $\infty$-cosmos of $\infty$-categories should already provide a sufficiently rich setting to re-do Lurie's approach in a straightforward way.
Alternatively, one could build it from the ground up as you seem to be suggesting. If you wanted to use dendroidal sets, say, this might be the way to go.
But I think it's an enticing open problem of the "low-hanging fruit" variety.