I actually think it’s extremely important from a character standpoint that Jon and Martin bicker so much throughout Season 5.
From the standpoint of Martin’s character: Martin is someone who’s always been entangled by The Lonely. When we first met him he was passive, and kind, and really just wants to be liked despite Jon’s irrational annoyance.
That’s why it’s so important that he learns to stand up for himself and advocate for his own feelings. Not just to anyone either, but to Jon, who he’s always wanted the approval of.
If Martin bit his tongue and just went along with everything he wanted, then it would show that he hasn’t grown a lot since the early seasons where he put “keeping the peace” over his own feelings. in this case his loud disagreement is a good thing because it means that he’s grown into someone who’s comfortable standing up for himself in his relationship.
From the standpoint of Jon’s character: Jon need’s someone to be willing to disagree with him in Season 5.
At the start of season 5, Jon has basically given up. He’s wasting away in the cabin, blaming himself for the end of the world and thinking that he can’t fix it. It’s Martin’s willingness to contradict Jon and go “I think there’s still hope. I think there’s still a world to be saved,” that pushes him to get out of bed.
Then during the apocalypse when they start disagreeing on how to handle saving the world and the merits of vengeance, Jon still needs to have those disagreements. Both because those new relationship hurdles are a reminder of the human connection he’s working to keep, but also because it forces him to consider a perspective that’s not his own.
The Eye is chipping away at his humanity, and everyone else in the apocalypse is treating him like he’s a God. He DESPERATELY needs someone to remind him “ Hey. You aren’t inherently right because you’re The Archivist. Let’s also consider other people’s points of view,” due the situation.
From a character standpoint I think a lot of Martin and Jon’s arguing is actually a really interesting showcase to where they both are as people by Season 5 and indictive of their character development. Also they’re in the apocalypse and that totally contributes.