The show might be about the doctor, but it's not as if the companion being the opening focus is a new thing.
Susan is the trigger to bring Ian and Barbara into the Doctor's world and is only the focus of the story for about 10 minutes or so. Then the Doctor comes on screen...
Ian and Barbara take the part of the audience identification figures. You're introduced to the Doctor's world through their eyes. And, because there are two of them, they discuss with each other their reactions to what's going on.
As people became more used to the Doctor, this wasn't so essential, and by the end of Season 2, with the departure of Ian and Barbara, the figures were gone for the next year. The next time you get these sort of figures is with Ben and Polly - especially in the Power of the Daleks, where they have to deal with the regenerated Doctor.
Once again, you get the companion meeting the third Doctor in Spearhead from Space: Liz Shaw is - for that story - our bridge into the world of the Doctor. This only really lasts for this serial, though. Sarah Jane Smith is probably the best audience identification figure to come around, and she keeps the role pretty much throughout her run. It's very obvious in the Time Warrior.
Looking at the "story of the companion", which is a key feature of the new series of Doctor Who. It's mostly absent from classic Doctor Who serials; the first real attempt at it came with Leela, but the change in production team gave us only one real story where her potential arc was explored (the classic Talons of Weng-Chiang). The next attempt was with Turlough, and we got a really great arc through his initial three stories (the Black Guardian trilogy: Mawdryn Undead, Terminus and Enlightenment), but the character was then basically forgotten about.
The first companion that really had a large focus of the story around them was Ace. It might not be on the level of the new series, but certainly with the Curse of Fenric, Ghost Light and Survival you have the Doctor actively seeking out things that have to do with his companion. (Or at least, dealing with).
The trouble I'm having with Clara at present is that she isn't the Clara I fell in love with - the Clara of "Asylum of the Daleks" and "The Snowmen". She's not done yet. I'm really looking forward to seeing where the series takes her.
Cheers!