There are only usually three bulletins a week on CNNI that come from that New York 'set' - two on a Friday evening and one on a Saturday (sometimes an extra Friday one if there is no State of America, and occasionally the business shows) plus CNN News Now. In that context I can understand why CNNI don't want to invest in anything more elaborate just for 90 minutes a week - and using a large CNN/US studio would probably increase staffing costs. There doesn't seem to be anything 'in between' this set up and a large studio at CNN New York - the studio used for Early Start/CNN US Newsroom seems bigger than the previous set up
From a studio perspective it would probably make sense to return this shift to Atlanta, where the studio is already in use at similar times for World Sport - but that would mean Zain Asher relocating back to Atlanta, or being found a new shift
(Ideally the New York CNNI Newsroom shifts would have more bulletins and that could help make the case for designing a better set)
Indeed that last sentence is probably the crux of the matter. CNN's news output at weekends isn't plentiful anyway but it's particularly pathetic on weekend evenings/overnights so there's no incentive to use a proper studio for the very sparse bulletins. However, the set up does look and come across as cheap especially on a global news channel.
I do think if they do want to use New York at the weekend using the Brooke Baldwin wall of the Newsroom/Early Start set and creating a Skyline look there like they had for Newsroom LA and have for the CNNI Atlanta bulletins would work. Those backdrops despite being similar in concept to the NY backdrop are far nicer on-screen and have much more depth.
Though as you say using Atlanta would probably make more sense or having London doing weekend daytime since there are a fair few anchors spare in London. With the 3 daily business shows from New York (First Move, The Express and QMB) and how often Quest is abroad filming shows like World of Wonders or Business Traveller there's plenty of business programming that Zain could cover in New York. Her energetic style too much more suits the business programming than news programming in my view.