What is the preferred method to roll out updates to hundreds of databases
I would adapt your fast solution to something more easily extensible, like PowerShell.

It would be trivial to have PS read a text file for a list of servers to run against, and another file or files for what commands to execute.

You could also add error handling and such like in your SSIS batch, and have it fire asynchronously so it would work on all your servers simultaneously.