Out of curiosity, the post you linked mentions that it won't work for renames. What's the approach for these and other types of procedural migrations, such as data transformations (ie: splitting a column, changing a type, etc.)
With a declarative model, would you run the migration and follow immediately with a one off script?
For both data migrations and renames, there isn't really a one-size-fits-all solution. That's actually true when doing data changes or renames in imperative (incremental) migrations tools too; they just don't acknowledge it, but at scale these operations aren't really viable. They inherently involve careful coordination alongside application deploys, which cannot be timed to occur at the exact same moment as the migration completion, and you need to prevent risk of user-facing errors or data corruption from intermediate/inconsistent state.
With row data migrations on large tables, there's also risk of long/slow transactions destroying prod DB performance due to MVCC impact (pile-up of old row versions). So at minimum you need to break up a large data change into smaller chunked transactions, and have application logic to account for these migrations being ongoing in the background in a non-atomic fashion.
That all said, to answer from a mechanical standpoint of "how do companies using declarative schema management also handle data migrations or renames":
At large scale, companies tend to implement custom/in-house data migration frameworks. Or for renames, they're often just outright banned, at least for any table with user-facing impact.
At smaller scale, yeah you can just pair a declarative tool for schema changes with an imperative migration tool for non-schema changes. They aren't really mutually exclusive. Some larger schema management systems handle both / multiple paradigms.
For MySQL/MariaDB with Skeema in particular, a few smaller-scale data migration approaches are discussed in a separate post, https://www.skeema.io/blog/2024/07/23/data-migrations-impera...