So, if we think of the reader as "what we have" and the writer as "what we want", then it's allowed to edit the writer because it's not a fixed schema. However, the reader is disabled because it's supposed to represent an actual dataset. If you could make changes, then things could easily get out of sync with reality.
The easiest solution is to use Readers > Import Feature Types. Then you import the feature types (tables) you want to the reader that you want. It's also likely to be a quicker solution than going through each feature type in turn and editing it.
Then just delete the original feature types that you no longer need - and click Yes when prompted to remove the reader (else you'll have a bunch of readers without any tables to read).
That's also the solution to avoid this position in the first place. Instead of using the Add Reader tool, just use Import Feature Type to add extra tables to the same reader.