It is the nature and culture of the airline that leads to filthy cockpits. Some yeras back starting with one airline I found that the cockpit crew could throw their rubbish on the floor and would then summon the flight attendants to the cockpit to clean up. The first thing I said to my copilot who having just cleaned his headset with his cleaning/disinfectant towel and put the offending towel on the central pedestal was "Put that somewhere else such as the ash tray or wherever you wish, but not in our common space." -(that is until it ready to be cleared away). He was offended but I made my point. As a Captain one of the roles I have to take is to tell copilots that at the end of each flight they must ensure that the cockpit is clean and all rubbish put in a plastic or sick bag for the cabib or cleaning crew to clear away. Instructors and all cockpict occupants have a role to play in this. Life is not just about PTS it is the way we interact with other people as well (CRM) that make for a great flying career. If I get into an aeroplane after maintenance and the cockpit is dirty because of maintenance work then who ever the crew chief is, must arrange for the cockpit to be cleaned. The same goes for a crew change only.
By the way in any outfit, fit to be called an airline a professional Cabin crew will clear all waste within reason as it a hazard in case of of a fire. Basic logic.