Having a class at 50 or 60 has nothing to do with the armoury bonus, which only looks at whether you have a class higher than your current one, not at whether that higher class has reached any particular threshold. The real reason for focusing on the MSQ first is (as you mentioned later) that you'd need to go back and do it anyway if you didn't start there.
The whole game is structured around its storyline. All the level 50 and above content is gated behind finishing the original ARR MSQ. All Heavensward content is gated behind finishing the additional MSQ added in the 2.x patches. Then a lot of the level 60 content is gated behind the Heavensward MSQ (though I'm not sure if it's everything, like it is with level 50). And when Stormblood comes out, it will be gated behind the Heavensward and 3.x MSQ.
So, focus on the Main Scenario and any sidequests that have the plus sign in the bottom corner of the icon indicating they unlock features or content. Whenever those don't give you enough XP to continue, look to other XP sources, which can be Palace of the Dead, additional dungeon runs, or whatever. (Once a day, leveling roulette is really good for that, and if you're doing that with your main job it should provide all the additional XP you'd need.)
