Also, most satellites are intentionally as dumb as possible, just a "bent pipe" transponder, putting all the complexity on the ground stations which are easier to service if something goes wrong. There might not be much to do on the satellite itself.
All geostationary satellites need to be capable of at least some station-keeping to correct for drift, move them to other service areas, or move them to a graveyard orbit at their end of life. (Unlike LEO, GEO satellites don't carry enough fuel for de-orbiting, and friction is essentially nonexistent at that altitude.)
That layer of commands is hopefully very well protected.
Typically some form of HMAC authentication. You can read about it in the CCSDS Blue Book.
If your objective is to disable the devices like they've done, attacking the "easy" layer is enough so why waste time on unnecessary complexity? Of course they might well have also done recon on the satellite side and collected valuable data they can use in the next round.
The current generation of satellites themselves generally do nothing to the data stream - for each of the dozens of spot beams they're transmitting, they generally just take an RF signal from the ground station (multiplexed in various ways up to the satellite) and convert the frequency. Same with the receive path just in reverse.
The actual modulation/demodulation all happens at the ground station. This is because they expect modem technology will improve, but the satellite has to be able to work for 25-30 years. (Though in the industry they are talking about putting more and more 'software defined' functionality on the satellites, but again this will mostly have to go via their secure systems at the ground station, not from the terminals)
So there's basically no way to interact with any 'satellite firmware' unless you're in a very specific location (near their ground station) with extremely specialised gear.
Also, the affected ground stations are in Germany, the satellite belongs to a US company.