The only starter I’ve given away was to my younger sister, and that was simply because I wanted her to have her pick of any viable pokémon in the National Dex and knew I could secure a starter for her on time. Would you believe it, though? Hundreds of pokémon, and she chose squirtle. That isn’t a complaint, of course; I merely find it curious that our older sister started with charmander, I started with bulbasaur, and she rounded out the traditional Kantonian trio.
In any case, while I could serve as a starter point for trainers in and around Cerulean City, it’s just that the Sea Cottage is too remote to be convenient for them to travel to me. Likewise, I could always travel to them, as the regional professors of certain other regions do (Rowan and Sycamore, for example), but quite honestly, the children of Cerulean already have Nurse Joy.
The Pokédex According to Bill 