The notion that Kirk or Spock always have to save the day (or Scotty if it's a ship problem) is one of the things that dates TOS. However, I also don't like it when they have to 'force' characters to save the day e.g. Chekov is suddenly a transporter whiz and has to run all the way from the bridge to perform a task that really should be within the capabilities of any trained officer etc.
I do prefer it when a character is allowed to perform well in their chosen field instead of gifting that to someone else e.g. Spock carrying out code-breaking and communications repairs in TOS; Chapel assisting instead of leading bio-research etc. I suppose I also have to concede that if the characters never do anything outside that field they'll stagnate.
I hope the comic can maintain a better balance overall.
Having Kirk or Spock resolve the main problem doesn't date TOS — it's good storytelling, plain and simple. You want the lead character to resolve the story's problem, take action. Or in an ensemble, you want to person who is the focus of the story resolve the problem and take action. For example, HILL STREET BLUES. You want a character, say Furillo, to resolve the problem in his particular storyline while you'd want Detective Belker to resolve the one in his.
There was potential to have two storylines with Kirk and Sulu, where Kirk takes action on the overall plot of the Klingon conflict while Sulu has to resolve the subplot of his sister's loyalty. But neither Kirk nor Sulu act, they react from beginning to end. Honestly, in retrospect, this arc feels more like a TNG episode than a TOS one.