There doesn't need to be a sickout...
There doesn't need to be a sickout, just don't go to work when you are not fit or not rested to safely complete the duty. The sick rate will be greater than 3 - 4 times per annum.
How often have you flown single pilot, with the other pilot asleep in the seat? Followed Controlled Rest procedures? Submitted ASR-F? If you need controlled rest you should have signed on. If you plan on using controlled rest you are breaking the regs.
If you aren't able to safely complete the duty, don't do it. If it is due to insufficient rest, tell Crew Control and ask if there is a suitable duty. Even if you are on reserve you don't have to accept whatever they throw at you. File an ASR-F.
We have enabled the company's piss poor management for too long. Time to be a rational, responsible professional. Its not hard. The company doesn't give a !!!!, it is time for us to step up and be professionals about it all.