I don't think its any more believable for Chekov to run the Science station in a crisis situation, a job usually performed by the Chief Science Officer, than it is for him to step in as Chief Engineer. Though in both cases they hand wave it by saying the Chekovs have been studying under Spock and ScottyIn TOS Chekov filled in for Spock quite a bit.(Pretty much anytime Spock was unavailable) Surely there were junior Science officers on board, yet the job always went to Chekov. The reason being if you're gonna have someone spouting dialogue, use the guy already on the payroll.
Then set up your characters to make it believable. Every crewman can use a scanner. It's part of basic training - you don't have to be a science officer. Every crewman can perform basic engineering. It's part of basic training, you don't have to be a fully qualified engineer.
However, not every crewman can be a department head. It requires experience, administration skills, and rank, since some of the crewman who will be sub-ordinate to you will be Lts and Lt-commanders.
Make Chekov an engineering liaison but don't make an 18-year old ensign who qualified out of Starfleet a year ago, chief engineer. That's just plain daft.
Its a conceit of the media, not "bad writing"
Using a science station is not the function of the chief science officer any more than using the transporter is the job of the chief engineer.