@jeff_straterI shared the project using File > Share > Public Link, now I realise I'm supposed to export and attach the file (see below) since "Allow item to be downloaded" is not a feature of the free version (off-topic but personally I think this is a rather absurd limitation since files can be exported and shared using Dropbox, email or whatever anyway).
I understand what you say about Capture Position Pending state, but IMO it doesn't look like that's the problem here, since so much as clicking the assembly causes the previously assembled parts to jump to where they are supposed to be. Furthermore, once two components are assembled, any previous Capture Position operations that affect those components can be deleted to clean up the timeline without consequences (unless the joint is deleted or disabled, which is not the case here).
While we are at it, you may notice the attached file is at version number 15. Essentially none of the previous 15 save operations are worthy of being a version. They were all intended as regular "save" operations just in case Fusion 360 might crash. There really needs to be a way to have a "working version", stored on the server in this case, that is overwritten when Ctrl+S is pressed. When the user decides the design has progressed to a point that it is ready to be considered a version, he/she can commit a version to the server. Having tens of versions that are automatically created whenever it is very hard to locate the version you intentionally saved and name among an endless littering of versions named "User Saved". Just saying.
Fusion
