Yeah, I think the TOS and DSC Enterprises both being made as drawings (oddly, matching the style of the ones in the NX-01 ready room, which I always assumed were done by Archer in-universe, and were definitely made by someone hundreds of years before there was an Enterprise-D or -E to add to the set) and gold models is more to give options to the set decorators/producers about which one they wanted to use, rather than an actual intention to have both displayed side-by-side.
Even PIC has gone back-and-forth a little already, using the DSC Enterprise in season 1, but the TOS-era Stargazer model in season 2 was designed as a TOS-style ship, not a DSC-style one.