Completion time would probably end up creating the scenario they want to avoid, which is a strict meta of jobs that can complete the fastest on top of the general animosity that can come from people underperforming to fit the ideal completion time. It's not really a fun metric to fulfill, either.

I don't inherently think trash should be removed for a boss gauntlet only, though the focus of the difficulty in the dungeons should be from the bosses being reiterated on and not the trash packs. Trash packs between bosses might be able to serve as a good cooldown period from the prior engagement. But, I think that'd be fun so long as completion is the ultimate victory and not mired around with sub-completions that'll railroad the content to center around a handful of the best jobs within a four man environment, instead of letting people go on their favorites of each role. Since, ultimately, the mode should be designed with the idea of fun first and foremost.

Cosmetic rewards would be about all I'd want to see, it'd be rough if it ended up becoming mandatory pre-raid content to complete to have entry-level BiS for the new raid tier if you plan to do it week 1, especially since some raiders might not be interested and then feel forced/compelled to do content they dislike just to be moderately competitive in content they do like.