Why is this not a common use case?
Linode, DigitalOcean, Vultr, EC2/Lightsail and many many other companies lease dedicated servers and VPS. Lease an app server, lease a DB server, move the data and port the code.
Yes, no doubt there's some other 'devil in the details', but if it's already "old school legacy" architecture, why not just replicate that some place else that offers better service (and likely better value for money)?