joevitus said: "Pretty sure Equity requires directors to hold auditions, just as businesses must interview outside candidates."
Yes, they must hold EPAs (Equity Principal Auditions) and ECCs (Equity Chorus Calls), but any names they were seeing for Eliza, for both the original cast and the replacement, would have been by appointment, which is not required.
There are many reasons to hold auditions for a role like that, even if the director is positive that they want one specific person. Unless the production is being built around the star (such as the upcoming The Music Man with Hugh Jackman) then you can't count on anyone until a deal is reached and a contract is signed. In the meantime, you better have other choices ready to make offers to as well.
Additionally, just because a director may want somebody it doesn't mean that the entire creative team or the producers are on board with the idea. In that case, the best way to convince the rest of your team that your choice is the best choice is to not only have them audition, but to also see other options as well.
As for the direction of the replacement, that is entirely up to the creative team. Some may want carbon copies of the person they are replacing, others will be more open to looking at the piece again and giving the replacement an opportunity to make the role their own (this goes for understudies as well). There are no set rules for or against this (other than things like forbidding the use of teaching a role by making the replacement/understudy watch a video of the show).