I think it's like anything else in that regards; having a heavy focus on the plot isn't in itself a bad thing, but don't make that the main focus of the game unless you can find a fun way of doing so. I kind of think this is what inevitably killed the adventure genre. A lot of these games had great stories and characters or whatever, but when it all came down to it many of the puzzles ended up being illogical or just unituitive. Add in very similair gameplay across the genre and everything just got stale I guess. Grim Fandango excluded because that game ruled.
In regards to Trauma Center, it seems the exact opposite. You have a fun game here, why the fuck are you ruining it with this melodramatic anime bullshit? Are there seriously people out there who are like "I need ten min. of shitty cutscenes talking about the two dimensional feelings of the main characters between every 4-5 min. section of gameplay!" When you start a new mission you're not given the option to go directly to the operation, but luckily can press either start or select to have the dialogue automaticall scroll at a faster pace. While this is a good thing, it can still take a min or so per intro and exit from all gameplay segments. Throw in maybe 30-40 seconds of little 'cutscenes' in the middle of some operations and keep in mind that 99% of all operations have a 5 min timer that you shouldn't even come close to hitting. If you're trying to skip through, you're still spending 30-40% of the time in game not playing shit. That's retarded.
Here's an idea for the plot of Trauma Center: You're a doctor in a hospital and patients keep coming in that you have to fix. If you fuck up they could very well die. THE END.