What you describe sounds to me like it should work, although others may be able to discern the issue. If you post screenshots of the column properties for the 2 columns, someone may be to able to identify a detail that you’re overlooking. For instance, ensure that the values from column A that may need to populate column B are indeed included among the valid values defined for column B.
You should add a screenshot of the Valid if and Initial value properties for column B to confirm whether there are any issues there.
Also, in case you didn’t realize, you could consider creating a Subsites table (with columns for ID, Site, and Subsite) and using the Dependent drop-downs technique for both column A and column B.