As the title says, I think a lot of the pursuit quests are redundant in the sense that they need to be repeated on each character.
The Vow catalyst quest IS cross character. The progress updates across all characters. This is convenient to stop repetitive actions and overall boredom/frustration from having to repeat the same quest, multiple times, across all 3 characters.
An easy counter argument could be "Well, just abandon the quests on the other characters that have been completed, problem solved." Well, not exactly. I don't like grabbing my dailies from banshee and seeing the MIDA quest just sitting there, it bugs me. I see it, and I HAVE to complete it, but I've already done it on another character, I already have the multi-tool. I don't want it to sit there, but I don't want to do it again.
My current issue is that I'm working on the mystery box on my hunter, but now I would like to play on my warlock. The progress for that quest (and many others) cannot be continued on the other character. If there is a reason for this type of individual character quest progression, i think the player base deserves an in depth explanation. Because currently, it's quite annoying that the quest cannot be continued just because we would like to switch characters for different play styles or whatever other reason.
Almost all crucible quests are cross character, Along with gambit. Some PvE are cross character, but not all and i think this needs a second look from the developers. This is an excellent game and i would like to see it excel and continue to thrive. I believe one way to better the quality of life in this game is to remove the repetitive actions.