> often you have to inject data into new tables or columns

No tool can help you with that, simply because this kind of data migration depends on your particular business logic that the tool has no way of knowing about.

While SQL Server Data Tools has its warts, it has been immensely useful for us in making sure every little detail gets handled during migration. That doesn't usually mean that it can do the entire migration itself - we do the manual adjustments to the base tables that SSDT cannot do on its own, and then let it handle the rest, which in our case is mostly about indexes, views, functions and stored procedures.

After all that, SSDT can compare the resulting database with the "desired" database, and reliably flag any differences, preventing schema drift.