Ash did have the choice of choosing from the starter trio. But he wasn't able to, since he arrived late. The starter trio had been chosen by other trainers and Professor Oak gave Ash a Pikachu.
This plot was shaped in a clever manner to bring Pikachu up as Pokemon that will represent the Pokemon series.
Like you've mentioned, the game versions give Pikachu a minor role but it was enough to gain popularity among the players. An interview with the creator of Pokemon, Satoshi Tajiri, states so:
TIME: Pikachu is sort of marginal in the game. But it's now the best-known character. How'd that happen?
Tajiri: When they did the anime, they wanted a specific character to focus on. Pikachu was relatively popular compared with the others and potentially both boys and girls would like it. They heard a lot of opinions about this. It wasn't my idea.
The youngsters who had played the Pokemon games were attracted to Pikachu. The reasons to why Pikachu, out of all the other Pokemon, could be because of its characteristics. Ikue Otani (voice of Pikachu) responded in a similar way when asked the same question:
Why do you think Pikachu is so popular with young people?
IO:I think it’s like being the owner of a pet dog; you’re always wondering what your dog is thinking, but you also believe that you understand your dog better than anyone else could ever hope to understand it. You can tell its thoughts just by looking at its face or how it is behaving. Whether it’s hungry, happy or sad. That’s exactly how Satoshi and Pikachu communicate. Because Pikachu can’t say anything other than its name, the audience has to think about what the “Pikachu” noises mean and learn to understand the character. Ultimately, I think kids feel like they are Pikachu’s owner.
The reason Pikachu was chosen to be the mascot, was because of its popularity. It was a brilliant marketing scheme, by taking advantage of the popularity of Pikachu from the games, and incorporating it into the anime and merchandise, sales had soared. The dynamic yet cute relationship between Ash and Pikachu was also another factor to bring up the sales.
TIME: How does that translate to the U.S.?
Tajiri: It's interesting, because in Japan, everybody goes for Pikachu. In the U.S., the characters Ash [Satoshi in Japan] and Pikachu are grouped together. American kids seem to like that. In America there are more products sold with Ash and Pikachu together, not just Pikachu alone. I think Americans actually understand the concept of Pokémon better than the Japanese. The Japanese focus on Pikachu, but what I think is important is the human aspect--you need Ash.