Apparently finds are close enough to us that we should be able "to follow up, to see, for example with the James Webb Space Telescope that we're going to launch [next] year, the atmospheres and also to look at bio-signatures, if there are any" !
As you say, the contrast between seeing (star + planet) during a transit, versus star alone, is enough to identify absorption of infrared light by some chemical species.
The simulations in the above paper basically show they can infer some chemical abundances (carbon, oxygen) and some atmospheric parameters (inversion layers) for Neptunes and Jupiters. But probably not for Earths, so not for the system in the OP. Even for big, thick atmospheres, these spectroscopic characterizations take a lot of observation time (like, days).
The race is indeed on to develop instruments and algorithms for this problem -- e.g., detecting CH4 or CO2 in an exo-Earth atmosphere. There's feverish activity in the Astronomy community around this, and post-JWST missions are being formulated to tackle the problem.
The concept you link is a "starshade". It's a sister concept to a "coronagraph" -- both use an occulting disk to block out the light from the host star, so that a non-transiting exoplanet can be observed.
One goal is to image exoplanets directly, and another is to gather spectra for characterization. One of two current mission studies doing studies of the two approaches is HabEx (http://www.jpl.nasa.gov/habex/). Obviously, the starshade is more complex/cumbersome/expensive.
Using one method or another, you have to achieve a contrast of about 1e10 between the star and the exoplanet. For every 1e10 photons that come in from the host star, your starshade/coronagraph has to let at most one get through to the detector.
It seems like the major limitation for discovery is getting enough observation time for whatever certain point in the sky that scientists are researching.
Why don't we launch a second copy of James Webb?
I encourage you to read his reply.
Supposedly he is right and NASA money is well spent, it still not implies it is efficiently spent. Furthermore, he seems to writes off aids as a net loss in his analogy, without considering the benefits of children not starving (to death). Which is especially weird with the follow up
We need more young men and women who choose science as a career
/devils advocate
I'm under the impression that R&D is a significant component of the cost in these sorts of "one off" science projects/devices and consequently, manufacturing, launching and operating a second one might not cost anywhere near as much as the first one.