Hi @BrettSanders,
I think the major hurdle with the workflow you're suggesting is there are a number of areas where things can go wrong, and all the steps in the process are achieved outside of Esri software. While it is absolutely true that a new but "identical" SQL instance could run the services, there are a number of caveats including using the same FQDN (or SPN from an OS-authentication perspective), the same security/users/passwords configured on the instance, and no other underlying data changes compared to the original machine/database.
With that being the case, it's difficult from an Esri Support perspective to recommend workflows that are not officially documented, even if the workflow could technically work if implemented correctly. If you were to push forward with this plan, I would recommend having a plan to revert to the original stack in case anything along the way goes awry, which is made difficult since in an AD environment/domain having the same computer name twice within a domain is not possible.
There is another option to update the registered data store connection with a new SDE file when a new instance is stood-up and the data/security migrated to that new instance. The documentation only mentions password updates, but it works for hostname changes as well (documentation enhancement is in the works already). This would be a band-aid to prevent immediately needing to republish all referenced services, but the Service Workspace information would not be updated within Server Manager so it is worth republishing when the opportunity arises.
Register your data with ArcGIS Server using Server Manager—ArcGIS Server | Documentation for ArcGIS Enterprise
https://enterprise.arcgis.com/en/server/latest/manage-data/windows/registering-your-data-with-arcgis...
Hope that helps,
Chris
-- Chris Pawlyszyn