Alas, there isn't a more consolidated strategy for this kind of multi-year upgrade, and you definitely don't want to manually run any of the schema upgrade scripts. The server upgrade process will take care of that and other necessary parts of keeping things in sync. Any sort of custom scripting or detailed upgrade assistance for your specific scenario would only be available with a premium subscription.

One other alternative, depending on what you have been using LabKey for/what kinds of data you want to move forward, is to install a brand new/empty instance of 24.7 (with embedded Tomcat). This new-installation is a simpler and even scripted install process, covered here. Then migrate work using folder archives or individual spreadsheet export from the old instance and import into the new one. Watching the logs at every step is always important to catch any issues, whether you go the database upgrade or old->new data migration route.

Hope this helps,

--Molly