In my head, the ideal process for analysing a story seems simple: start watching (or reading), take a note of any problem I see & keep watching. In practice it's one of the hardest skills to master.
For me (and, it seems, most groups) it's easy to forget about keeping my eye on the big picture, and instead get bogged down in details.* Instead of discriminating between big problems and little ones, I treat everything equally and try to solve them as soon as they come into view. The upshot is that I spend lots of energy on fixing inconsequential issues - and then my brain isn't fresh at the point I need it most; when I run into the big problems that always seem to lurk towards the end of a piece.
The solution: nominate someone to be the Note-taker. Your team starts analysing the script, working their way from A to Z. If you find yourself getting bogged down in detailed problem-solving, it's the Note-taker's responsibility to identify that. They should force you to move on (while making a clear note of the issue that needs to be resolved).**
*This problem has hit me in script assessments, readthroughs and (especially) in the editing room.
** In other words, it's a pretty clear application of the Blue Hat phase of de Bono's 6 Hat process.