Because his three flying types aren't all birds. They're very visually different, and have really distinct styles of 'flying'. Talonflame is the standard bird Pokemon he always gets, but its secondary fire typing really sets it apart. Noivern is a dragon, and Hawlucha is a fighting type who happens to have the ability to glide.
On the other hand, imagine having a Talonflame and a Swellow on the same team. Wouldn't it be much harder to set those two apart stylistically? Granted, Pikipek's evolution could have an interesting gimmick or secondary typing that would make it work, but idk I just think two birds on his team would be kinda redundant.