The simplest thing you can do in a scanner is have them sit there doing nothing while you acquire scans. In the end, you just have a 4d matrix of unsigned integers. For each voxel you can acquire an average over the scan and check whether it is significantly above zero using a t-test. Given enough data everything will be significantly greater than zero, including parts outside the head. Or you can compute a global mean to center all the voxles, and check which parts of the brain are significantly above the average, or significantly below the average. Extremely simple analyses (and so yes, there are lots more you could do).
In task fMRI, you have them do a task, and you use events in your task design as predictors of the BOLD, and then display a voxel map of either the beta values, or, more commonly, the T-values of those regression betas (or a contrast of those regression betas). In this case, you really aren't looking at activity. You are looking at correlations.
Those islands of activity in whole-brain analysis images in figures in papers happen because the result images are thresholded, e.g. at p < .05 false-discovery-rate correction for multiple comparisons. Personally, I think unthresholded images are better because they are more informative.
Let's take a concrete example. You have a subject do a task where they have to choose between two gambles varying in risk and reward. Then, for each voxel, you predict the BOLD time course using a series of events (time of presentation of gamble options) with magnitude equal to the coefficient of variation between the two gambles. So now, for each voxel you have a beta value showing how CoV predicts BOLD. You notice that anterior insula on both cases has the highest beta values. You threshold at conventional statistical signficance, after correcting for multiple comparions, and all the spurious, or less important, correlations drop out of the image, and you are left with two bright spots on a map pin-pointed on the left and right anterior insula. See: in this anaylysis, not all "psychologically and biolotically meaninful activity" is being examined or looked at. For example, button presses events show up localized in the motor areas too, but they weren't looking at those. But they could have, if we were interested.
As such it’s impossible to say what subset of brain activity is directly related to some activity rather than some related mental processes.
There's a lot of signal processing theory, regression analysis, etc. that goes in to it. With that said, the issue that arises is that only the strongest correlates may surface or be observed. There may be a lot of brain activity that overlaps or looks identical between that resting state (baseline) and the stimulation state, but can't really be included because a distinction can't be made, which goes back to your final point. That, however, doesn't mean that there aren't actual subsets of brain activity that we can attribute to a specific activity.
For example, if you're in an fMRI and I show you nothing but a fixation cross, then show you a sad image at a specific time for a specific interval, then went back to the fixation cross, and also showed you a scrambled version of that same image, I can (oversimplified) run a diff check to see what's different between the three. From there, I can remove the overlap between the three from normal baseline activity and activity specifically related to visual processing (scrambled image) to reveal what's specifically different with that sad image. If I see that the amygdala has much higher activity during that sad image compared to the fixation cross and the scrambled image, I can conclude that it has some role in emotional processing (well not really in this example because you'd also need a ton of images, including neutral images, with different orderings of each to really be confident; also really couldn't make that conclusion, just present the correlation because...science).
The point is that there are methods to isolate activity related to a specific activity/stimuli/response/etc. However, it's currently very difficult, if not impossible, to make distinctions in those overlapped areas.
Note: this is obviously an oversimplification of neuroimaging research and analysis
It’s expensive to run brain scanning studies, so studies have smaller populations or lower resolution data collection in response. Yes, brain scanning studies are new, but we’ve been doing fMRI studies for nearly 3 decades and it’s still expensive. Had that cost been scaled back, we’d have more and/or better data because grants are (very) finite.
> So why does the myth of a compartmentalized brain persist? One reason is that brain-scanning studies are expensive. As a compromise, typical studies include only enough scanning to show the strongest, most robust brain activity. These underpowered studies produce pretty pictures that appear to show little islands of activity in a calm-looking brain. But they miss plenty of other, less robust activity that may still be psychologically and biologically meaningful. In contrast, when studies are run with enough power, they show activity in the majority of the brain.
Another, more relevant reason, is that brain-scanning studies are in their infancy.