You're coming at this question the wrong way around IMO.
IF the goal is the give Broadway a more multi-purpose space, a much more elegant solution would be to take one of the existing multi-purpose spaces in the city and go through the process of making it an official Broadway house.
Gutting and reconfiguring a Broadway house is an expensive and labor-intensive process, and even after all that, you're still squeezing a square peg in a round hole. These proscenium houses weren't meant to be used like black-boxes. And I'd argue that the precedents we've seen have proven that (I wasn't a huge fan of the staging of Here Lies Love or even Great Comet on Bway, which might be a hot take).
On the other hand, getting an existing space qualified as a Broadway theatre is more of a bureaucratic process, which I can't imagine would be as difficult or expensive as gutting/reconfiguring a proscenium house. I guess the problem is, most of these spaces in NYC are run by non-profits who probably don't WANT to be considered Broadway houses (e.g. St. Ann's, The Armory, The Shed**, etc.). But maybe that will change.
Or maybe a warehouse-type space can be converted into a new theatre, with Broadway accreditation - still probably less expensive and more elegant than trying to give an existing Broadway house that degree of flexibility.
**(Interesting that you dismissed The Shed so quickly simply because it isn't well-known. Again, that seems like a much easier fix than a massive architectural overhaul. It's just a matter of marketing. And that's something they're in the process of doing anyway - it's a new-ish venue, still in the process of establishing itself in the theatre landscape, and they're making good progress with their high profile projects.)
And all of that is IF we assume that giving Broadway a multi-purpose space is a worthwhile goal. But I'd argue that core premise is still the wrong mindset. Rather than straining to contort these productions and theatres, why don't we start giving Off-Broadway the same respect and visibility that we give to Broadway? The Shed, St. Ann's, and the Armory are already providing exactly the service you're looking for. If Guys & Dolls is looking to come to NYC, they have the power to do just that, by transferring to one of the existing spaces that suits their needs. Why does it have to be Broadway to be seen as worthwhile?
