I’m going to say that there is no such thing as a best class. It’s just what fits your playstyle better. And I have to agree with KK that playing until level 5 is not a valid way of doing it since some characters don’t take off until you’ve finished the 1st playthrough. Others like Zer0 change as you get one pointers, forcing you to somewhat change your playstyle around them. Some are gear dependent, others, well not so much. Like Salvador needs very specific gear to achieve the levels of broken he’s known for, while you can give almost any gun to Axton or Maya and they can make it work no problem. As someone who plays every character, I’ll make a quick summary of them:
Axton: General all-rounder. No frills, but he gets the job done. Good with really any gun type.
Maya: Can immobilize enemies, spread slag like no tomorrow, great crowd control, great health tanking abilities. Like Axton, she can make almost any weapon work, but she’s weak at raiding.
Salvador: General powerhouse. Has two guns and screams a lot. Painful to play until you get specific gear, but broken powerful once you do.
Zer0: Glass Cannon. Has a potential DPS higher than Sal, but it takes a lot of work to reach that point. In my mind the hardest class to play as because he’s a be precise or die character. Deception and Kunai make him great with every gun type.
Gaige: Enjoys up close combat, DT is either a great supplement to your damage or a great distraction depending on how much you spec for him.
I know you said you don’t want to play Krieg, so I won’t bother writing one since I understand why some would not find his unorthodox playstyle enjoyable.