I've always found this very annoying, and I think I filed a feature request for this addition years ago. This is a user-to-user forum, so you should file a feature request (or search to see if there are any existing ones first) on the official AME UserVoice, then post a link here so can easily add their vote: https://adobe-video.uservoice.com/forums/911308-media-encoder
The workaround to this, though, is to browser to your project file in the Media Browser where you can expand the "Name" column's width.
Interesting because they should take the same time as both panels are spinning up the Dynamic Link Server. I actually have always felt the opposite of you and don't like waiting for that little dialog to populate and prefer the Media Browser. Either way, it's annoying that there's no way to expand the column, so if you file a feature request and post the link back here, I'll happily vote on it.