Gio is the consummate captain. Hardest working guy (fittest player year after year following Iggy's footsteps) plus he's a skilled offensive player. Undrafted he had to earn everything so he appeals to the grinders and the stars. Community work is award winning too.
Gio is also emotional on the ice and you see him getting after refs but he's never been had a reputation of a rat or dirty.
The media is an important part of the equation. They can drive narrative and media types decide who gets into the HOF etc. You have to be well spoken especially after tough moments. The media doesn't want to be looked down on despite the sometimes dumb questions. Lots of public interaction to the team is through media and media types. Gio is absolutely fantastic in media interactions.
Look at Rasmus here (I really like Rasmus and think he's really thoughtful and good at breaking down games). To me he can barely keep his contempt over the dumbness of some questions:
https://twitter.com/user/status/1360385701687160833
Could Rasmus be captain or wear a letter? Sure but he's got to be able to hide what he thinks about media questions (I might be wrong about his disdain)
So back to Tkachuk: why could he be captain? He's a smart, confident, skilled player who gets into the tough areas. He's articulate and drags his teammates into 'the battle' but he may have been a little to immature with some comments early in his career. At an already young age he wears a letter. His hockey family has exposed him to NHL life and that gives him an understanding of all aspects of being a professional hockey player (being a good team mate). I think he leads by example (sticks up for teammates on ice and off) and is vocal plus good with media.