Wanted to look at how some of the features affect damage.
Here's a chart of various Champion builds:
For now, this is just the Champion. The Battle Master chart mostly just bumps everything up a little bit.
Consistency
Any melee weapon you pick is going to have very similar damage profiles, as seen above.
Note: The maul doesn't have any weapon mastery modifications added. Topple is difficult to model. After level 9, you could change it to various other options, and Graze would actually be a nice buff in a comparison where the base accuracy is assumed to be 60%. Accounting for those should put it on par with or slightly above the PAM build.
The main exception is for ranged attacks. The longbow above illustrates that. This is largely due to the fact that there aren't really any damage-boosting feats for ranged attacks. At best you can use Great Weapon Master for the extra Proficiency Bonus damage using a heavy weapon. (The above chart does not include GWM for the longbow. It would provide a mild boost, but still be noticeably below the other weapons.)
The longbow is at roughly the same level as Treantmonk's baseline Eldritch Blast; maybe even a touch below it. On the other hand, being able to shoot from hundreds of feet away, and slowing (or pushing) the target at the same time, means it doesn't have to be super powerful to be effective.
Studied Attacks
I added Studied Attacks at level 13, but it's not as impactful as I would have expected. But that's partly because of the builds I'm using.
With Studied Attacks, if you're using a Vex weapon you pretty much have perma-advantage — either Vex gives you advantage when you hit, or Studied Attacks gives you advantage when you miss. More so on a Champion because you can use Heroic Advantage to ensure you still have advantage when switching targets.
However, because Vex already gives you advantage a lot of the time, the addition of Studied Attacks to the TWF and Rapier builds isn't very noticeable.
Meanwhile, the PAM build gives you Graze. Getting advantage raises your effective accuracy, which reduces how much Graze adds to your damage. That means you gain less from Studied Attacks than you might expect because it's cutting into the damage you would have gotten from Graze.
And for longbow, the Archery fighting style raises your accuracy, which reduces the value of Studied Attacks because you're not missing as often.
The maul gets the most out of Studied Attacks, of those listed. You can see it spike well above TWF at level 13, after being on par at level 12.