Suppressing pixel readings is not plausible with the current CCD designs, as far as I'm aware. These devices are so sensitive that any wiring needs to be done _extremely_ carefully to minimize crosstalk and keep temperatures extremely low (think under -100C).
Masking in data processing is the more likely technique.
Better than suppressing pixel reading would be suppressing pixel changes. The first process lets pixel to change as a result of incoming photons, the second one maintains the "unread" state.
Yes, current CCD devices can't do that. Advantage of this change is also in reducing crosstalk - if we need, say, to remove accumulated charge and for that we need to draw the current, the current may influence our reading of neighboring pixels. I'm tempted to look towards "smart mirrors" - MEMS systems, even for mirrors of 1 sq. micron - but purely electrical solutions might be possible.
This is functionally the same, but a less flexible technique, as reading the pixel values anyway, and then discarding or covering up the values during data analysis/preparation. IRAF already contains these tools.
Getting shorter exposure times is actually helpful to mitigating satellite contamination. In the last 5 years, we're down to 1-2 electron noise per read, which is a full order of magnitude improvement.
The streaky images in the press releases aren't meant to communicate "we don't know how to clean data," they're meant to communicate the severity of the interfering signal which must now be removed. Imperfectly, with imperfections in proportion to the severity of the interfering signal. Which is large.
Here is the second one. Earth observatories demonstrate streaks of satellites across astronomical images taken over comparatively long time. We know that at each moment satellite is in just one place in the sky; more, those constellations tend to have rather predictable orbits. Image sensing devices can use knowledge of satellite orbits to suppress reading a particular pixel at particular time if a satellite is expected to shine towards that pixel.
It's quite delicate task, I understand, but I also know that practical astronomy is known for inventing solutions to very diverse problems. This approach won't completely restore the abilities to observe the sky, but can significantly diminish the impact, discussed in the article. Especially if Earth-based devices would be constructed with such "shielding" (remember star shielding invention when working on exoplanetary discoveries?) in mind.