If it is just
perception as you say then why doesn't the company allow the pilots to make their point and have it disproved. This would give the company far more credibility than threatening them giving the
perception that the company has something to hide.
If the safety culture in a company prevents pilots from calling sick for fear of reprimand then there is something wrong with that companies safety culture. Having read the Colgon crash I have yet to find any reference to Breitlings and the like…that smacks of smoke and mirrors….

I have however read about pilots having to claim milk tokens and work second jobs because their salaries are so low that was how they had to survive.