No, this passage does not say that leprosy can be cured using birds. The rite described here is not about "curing" at all. Rather, it is about the ritual purification. These verses describe the first stage by which one previously excluded from the community was restored to fellowship within the nation of Israel. The (former) leper must have been healed (by whatever means, here unspecified) prior to the visitation by the priest, or the ritual will not go forward. This can be seen in verse 3:
and the priest shall go out to the outside of the camp. Thus the priest shall look, and if the infection of leprosy has been healed in the leper... (NRSV)
The verb "has been healed" translates a perfect passive form of רפא (rpʾ) "to heal". The syntax (literally, "behold, he has been healed") clearly indicates that the cure was completed prior to the encounter with the priest.
Note also how verse 7 and following refer to the individual with respect to the bird procedure:
he who is to be cleansed (המטהר)
The verb use here – טהר (ṭhr) – generally refers to instatement or declaration of ceremonial purity. Here it describes a separate event from the healing which has already taken place.
Gordon Wenham's comments on Leviticus 14 drive home the point:
The procedures described in this chapter are not curative but ritual. The priests did not do anything to cure the sick person. Their duty was to diagnose when a man was unclean and when he was clean again, and to make sure that the correct rituals were carried out when the disease cleared up and the man was readmitted to the community. To use a modern analogy, the priest in ancient Israel was more like a public health inspector than a physician. He determined whether a person was infected; he did not attempt to cure him.
Gordon J. Wenham, The Book of Leviticus, NICOT (Eerdmans, 1979), p. 207.