It was not Sheldon but David, the poet's friend, who illustrated his new poem, published in the journal.
Can we remove the who? as the parenthesis itself is a who clause.
No. You can't. Reduced relative clauses are applied to relative clauses with
EITHER “simple present tense”:
... poet's friend, who illustrates his new book => ... poet's friend, illustrating his new book
OR “passive perfect or past”:
... poet's friend's new book, which was (has been) illustrated recently => ... poet's friend's new book, illustrated recently
and a few more derivatives of these cases . i.e.: “passive present continuous”:
... poet's friend's new book, which is being illustrated => ... poet's friend's new book, being illustrated