The dynamics of the original cast of the show were actually surprisingly well-designed. Each of the four core characters had a fairly distinct narrative function, and it never felt like they were redundant to one another. Unlike in the later Hartnell era (and arguably the rest of the series...) where companions became fairly interchangeable devices for screaming and asking questions.
I believe the only other time there was a four-person TARDIS crew was during part of the Davison era, and in that case it just seemed unwieldy for the writers, since there weren't really enough distinct things for all four people to do for much of the time.
Given that, it will be interesting to see how the upcoming season with Whitaker as the Doctor works out... I believe the press has been promoting a four-person main cast for this season once again.
You really have to give credit to the original production team. Like you said, everyone in the TARDIS at the start of the program serves at least one vital function, even if sometimes it degenerated into screaming. ;)
I'm pretty sure you're right about Davison having the only other four-person TARDIS crew, since at one point he toted around Nyssa, Teagan, and Adric. Later on, after Adric's death and Nyssa's departure, he picked up both Turlough and Kamelion, making a second round of 4 companions, though Kamelion only made two on-screen appearances, I still say he counts. :)
Ohh, that's right; I forgot about Kamelion! Yes, the four-person section of the Davison era was even longer than I was thinking. It would actually include Planet of Fire as well (Turlough, Peri, and Kamelion). And Planet of Fire is actually another good example of one where all three companions had something reasonably interesting to do.
Kamelion is actually an interesting case, because even though he was a companion in principle, his only two appearances had him being brain-dominated by the Master and effectively functioning as an antagonist. It would be really interesting to see how the writers would have used him as a proper companion. Maybe Big Finish did some programs from that era I should investigate.
The reason Kamelion saw so little action is really sad. The man who invented and controlled the prop for the show died unexpectedly. Unfortunately he passed away without leaving behind any documentation about how the robot worked, so the production staff neither knew what to do with him nor how to do it, and it was decided to retire the character. That's why Kamelion doesn't even get a mention or appearance in The Five Doctors, despite him having literally joined the TARDIS crew in the previous episode. :(
Kamelion's gotten a lot of attention and character buffing in the Big Finish audio productions, as well as some of the Virgin Past Doctor adventures, and even the comic books. Some of it's pretty cool, some of it's extremely weird (Kamelion and the TARDIS at one point have a child together, which Peri has to care for--WTF?!), but Big Finish and Virgin have both been great at filling out back-stories for minor characters. Turlough features in a number of them as well, and gets a nice fleshing out too of his time before the Black Guardian gets his claws into him. :)