I cross-referenced the definition of bird with the sprite images on our Pokedex here on the DB, so here are "Bird" Pokemon that aren't part-flying.
bird (bərd) - noun - A warm-blooded egg-laying vertebrate distinguished by the possession of feathers, wings, and a beak and (typically) by being able to fly. Source.
Blaziken (Resembles a bipedal chicken with a beak and "hairlike" feathers)
Latias (I explain this towards the bottom of the answer)
Latios (I explain this towards the bottom of the answer)
Latios and Latias: How are they birds?
Latios and Latias are both "combinations of a jet plane and a dragon" (Source). Cross-referencing these Pokemon with the definition of birds, we can tell Latios has feathers on its tail and Latias from its PokeDex entry:
Latias is highly sensitive to the emotions of people. If it senses any hostility, this Pokémon ruffles the feathers all over its body and cries shrilly to intimidate the foe. (Source)
They both also have wings that they can contract to fly as fast as a jet, and they both have beaks that end up coming to a sharp point on their nose. They must also be able to lay eggs because there is more than one Latios and more than one Latias across all six regions. They are in Kanto, Hoenn and Unova, as seen by their locations in the games. There is also a pre-alpha picture that supports this:
This concept ended up being split into both Blaziken and [email protected], and supports that Latios and Latias are bird Pokemon. Since this is a shared concept, Junglefowl (believed ancestors to the chicken) could have possibly influenced the Eon duo's design.
Source: Above Links