I never noticed Jesus refers to birds as an example of worry, or lack thereof, in Matthew.
In Matthew 6 Jesus advises us to not be anxious about things. He points to the “birds of the air,” who neither sow nor reap, but still don’t get all ado about nothing [v. 26].
Four chapters later, Jesus shares that we do not have to fear those who are out to do us harm. He again points to birds, sparrows specifically, as our example [v. 29]. They are as cheap as cheap can be, nevertheless, God is taking care of them as if they were Blue-billed Curassows (#2 on the endangered bird list). Not one of them will bite the dust unless God says so.
The birds show us not to be anxious, they teach us not to fear.