To me, the companions have a little bit of that old horror stupidity, you know where you're shouting at the screen "Don't answer the door!" And they answer the door and get killed. I suppose this is needed to offset the Doctor's brilliance, but I could do without.
As I understand it, originally the companions were there to create a connection to the viewer. They have to have a bit of naivete to create that connection.
It's another reminder that Doctor Who, despite its setting, is not truly science fiction in the technical sense; like Star Wars, and to a certain extent Star Trek, it is grand fantasy that happens to include space travel, time travel, and aliens. The Doctor takes
Clarke's Third Law to extremes. He is a demigod by any reasonable standard; the Doctor (and other Time Lords) are not merely brilliant but exceedingly powerful. The Doctor and the Master compare quite well to Gandalf and Sauron in terms of raw power (and the ways in which each uses their power).
(Digreesion: One could reasonably say that Davros's statement about the Doctor turning innocents into weapons applies pretty well to Gandalf. Frodo and the other hobbits are manipulated by him into taking on the quest of the Ring, and the other members of the Fellowship were on some level pawns moved into place by Gandalf--Aragorn and Gimli to a greater extent, Boromir and Legolas to a lesser extent.)
Anyway... against the backdrop of such (literally) fantastic power, the viewer needs a person who is no less heroic but much less powerful and, well, naive, human, hopeful.
I do think the romantic subplots with Rose and Martha got a bit out of hand; I was worried about that with Amy, but the inclusion of Rory made that unlikely. (I have yet to see the Clara episodes, as I'm catching up with all of this on Netflix.
TLDR: The Doctor is a demigod, and the companions necessarily seem weak in power, though not weak in spirit, by comparison.