I thought you had to get to the end of his quest within the time limit to get him? Is that not the case?
You don't actually gain anything from completing Clive's quest. All you really get is a little bit more of Clive's story out in the open.
But yeah, Elza is actually somewhat easy to miss if you aren't looking around Muse or paying attention since she's tucked away in the northeast corner of the town where there aren't any shops (outside of a Blacksmith that isn't really worth a drat IIRC) or other major points of interest besides her. She'd be a dick move if she didn't have a custom sprite and something like a Sound Set or a Window that you can get off of her.
It's also not so obvious to some players that her package is what you need to get Clive since you either already have the package when you meet him and get him, or you don't and thus can't get him at all. And the package itself is treated with such little fanfare that subsequent playthroughs are more likely to miss him due to thinking that you only need to talk to him to get him to join (especially if said player is trying to skip over everything to complete his quest in the first place).