Birth by Sleep Should Terra have become a Keyblade Master?

Probably not, no. Sure Terra always did what he did for the right reasons, but the how of what he did was usually the wrong way to go about it.

Let's start by looking at his fight with Eraqus. Terra wasn't just trying to stop Eraqus from hurting Ven, he was trying to kill the man he thought of as a father. Notice the way he acts in the cutscene after the fight, he behaves like he's lost himself in anger, and he clearly regrets it. He had very good reasons for being mad, but he shouldn't have let his temper rule him like that.

To address the Riku vs Terra argument, look at their different darknesses and their behavior when using the darkness. Terra's darkness has more of an inky look to it, it flows out of him like liquid, and it's sort of a black-purple mix, like the Heartless. Early on in the series Rik's darkness had this inky look too, but as the series has gone on it's gotten a more solid look, by 3D it stays close to his body, tightly packed in whatever area he formed it in, or solidly in one shape, and it's either a flat black or black with white streaks. As for their behavior, Riku's doesn't change when he's using darkness, Terra's does. Terra pretty clearly becomes more aggressive when he's using the darkness (look at his dark impulse command style and darkness based attacks). Basically while Riku has control over his darkness, Terra's almost being controlled by his.

Plus Terra usually shows pretty poor judgment. Look how easily he was fooled by Captain Hook. Captain Hook. One of the most obviously evil characters in all of Disney, and Terra just takes him at his word. Actually look just how often Terra was hanging around the villain of whatever world he was in, and all they had to do to get his help was tell him that they were the good guy, even if the evidence suggested otherwise (Jumba in a jail cell, the evil queen asking him to assassinate someone for him, etc.). Even Sora, all-loving, friend to everyone Sora, can usually tell who the villain is as soon as he meets them. Terra though generally wound up as their pawn before he learned that they were actually bad guys.

And the way he reacted when Aqua questioned him in Radiant Garden? Run off like an annoyed teenager.

Actually that pretty much sums it up.

tl;dr Terra acts too much like an inexperienced and immature teenager to have earned the mark of mastery.