Well when it comes to that Eddie is just beeps and boops, it could all just be programming. Yes Man is also just a series of programming being run. Its personality is no different than any other robot you come across, it just has more responses than "is enemy or not, if enemy then kill" and a few more variables that it can take into consideration. And when it comes to The Sink, I don't even consider it canon because of how stupid the entire thing is. One of the things being lightswitches with personality modules.
ED-E has more personality in those beeps and boops than most robots in games.
Also, Yes Man programing is one of the things that supports being able to play as a robot in a fallout game. Just have a robot with a specific personality programed in it. A learning/growing program (like Skynet from FO2). That becomes more efficient with experience.
Want to look at a not really AI robot, but with personality? HK-47 from SW: Knights of the Old Republic.
But it could be a robot like Skynet from Fallout 2. It would be interesting if we could pick what type of brain it has installed by picking a "Brain Trait" at the start, which would give bonus and penalties.
There's also the ACE computer from Fallout 2, that also proves that it's possible to have a conscious machine without true AI:
I am ACE, an Artificial Conscious Entity. I am more than machine but not as highly developed as a true artificial intelligence.
There's also Fallout Tactics Calculator, as another case of a personality, but not a true AI.
And like I mentioned, Fallout Tactics has a robot recruit (actually 2, but one is only for multiplayer), HR 1205. A fully controllable robot character. And the already talked about Skynet follower. So it's not unheard of robot characters in older Fallout games.