Okay, let's try this again:
I'm not going to lie when I say that the idea of a journey fic starting after the main character gets his/her Pokemon is a pretty innovative idea. It could lead to a twist about the character's past and then some. Though I might not do it, I'd like to see how someone else who's clearly better than me at writing try and execute it.
That said, I do believe that authors can write a good journey fic starting with the Main Character getting his/her starter Pokemon. I agree with Astinus that authors tend to start the main plot around the first gym after a few chapters but just because that tends to happen, it doesn't mean an author can't start with that beginning mix things up for his own journey fic to make things seem less obligatory.
As I said before, journey fics can start the main plot with the evil team of the region making it's first move while the Main Character gets his/her Pokemon but as you said, Jax, starting with that and wasting chapters on obligatory events alone is not good. The way I see it, if the author is adapting from the games, he/she can expand the role of the region's evil team to feature them more with her own stories to keep the readers coming. If he/she is writing his/her original journey fic, then that's easier. The writer can have the evil teams do more and have plot-related events occur before the first Gym that way. Whether they'll be good or bad is entirely subjective.
Another way to make the beginning seem less generic (you know, Trainer waking up and rushing to the lab) is know what makes it generic and do things differently to subvert those common tropes. Suppose the Main Character is a girl with a sense of fashion, you could start with her at a clothes shop getting a new outfit for her journey at the last minute. And rather than her rushing, she takes her time walking as she's not late. As for the evil team attacking the lab, the author can foreshadow things. Like say the story is in Hoenn, the author could have Professor Birch and another guy look at something related to Groudon and Kyogre, prompting Team Magma and maybe Aqua to attack the lab. When the Main Character and her/his rival arrive at the lab under siege, they can help by using a starter each to assist in repealing the criminals along with Professor Birch.
That was just an example of what I think how a typical journey fic beginning can be improved to make it atypical, so to speak. But that was just me alone, of course.
As for filler in fanfiction, I don't like it when it's clearly meant to pad out the story but I can like it when it's a side-story that the author thought up and wrote because he/she was inspired at the time and wanted to write a different story separate from the main story. I agree that filler that's essential padding isn't good at all for fanfiction but side-stories I can like.
That's all I got for now.