I would agree with David that have Shepard do research in the third game would not have been a great idea. Shepard is a soldier and it has been well established that his/her talents do not lie in research.
However, I strongly disagree with the generalisation that has been made at least in the OP. Science and scientific methodology can make for great drama and it's application can be visualised really well. I recommend watching the great Canadian tv series
ReGenesis as case in point (especially the first two seasons). Of course, things are over-simplified and exaggerated but that is just part of the show business (soldiers aver exaggerated in war movies, docters are exaggerated in hospital tv shows, etc.). As someone working in the field myself, I think this is a very good example of how science can work in a visual medium to tell a dramatic story.
Would this have worked for Mass Effect 3? Obviously not. First, as said above, it's not Shepard who would do the science. Secondly, while this works well in a tv show or film, it would be tough to come up with a game mechanic that is interesting long term (and I think we all remember
Shepard's little venture into biochemistry in ME1

).
However, this does not mean that research wouldn't have worked in the plot of the game. It could have been Shepard's goal to support a research effort, organise supplies, gather intel and new specimens (read: reapers), not to mention to keep the research facility safe. There are even hints of this in the actual game as you gather war assets that help the crucible, etc.
I do think the plot would have benefited from giving our cycle a bit more agency in the fight against the reapers and not - yet again - relying on some miracle machine from the past. At the end of Retribution, Anderson and Sanders even talk about building a research unit to find a way to fight the reapers. I was rather disappointed that we never heard of it again in ME3.
All that said, I also believe that given the way the trilogy was (not) layed out, starting the effort at the beginning of ME3, when the reapers are already there would have been too late. This research theme would have been the perfect plot for ME2 in my opinion. I mean, in terms of the story, whoever actually needed the collectors? Having Shepard and research team discover a way to defeat the reapers in ME2 and then struggle to implement the plan in ME3, that would have been the way to go for a coherent trilogy. Ah well, too late now.
And just on my way out,
here is another way to show science in a visual medium.