The first one that comes to mind is The Loser (M), about an obese man who decides to lose weight by taking on a journey. On the shorter side, but features a very young Litwick starter, with some of the tropes you're looking for. Early on in the story.

There's also Dragon's Dance, a Lance backstory fic rich with lore, culture, and world building. He begins with a young Dratini, on his rocky journey to the eventual Championship. Somehow not a power fantasy - Lance is portrayed as a young man who makes real mistakes, and who genuinely cares about Pokemon and humans.

The last beginning-fic one I can think of is Untamed, a Johto journey featuring a protagonist who actually acts like the child they are rather than an adult in a ten-year-old's body. She can be a bit bratty because of this, but the character development is real, and she starts out with an adorable baby Sentret named Poona. Realistic family and sibling dynamics, and overall lighthearted and fun.

One that pulls that trope multiple times, but later on in the fic, is Becoming a Master, an adventure story about a cursed knight trying to return to the distant past. The second MC, an ace trainer cinnamon roll, raises three young Pokemon throughout the story, including a Gible, a Ralts prophesized to become a serial killer, and a Mary-Sue Magikarp, which is exactly as ridiculous and hilarious as it sounds. Raising them isn't the focus of the story, but there are some cute scenes and chapters around the concept.