Your thoughts are inline with the Church. Among the Eastern Catholics the practice is to baptize, give communion and confirm (chrisamate) the infant. Infants and very young children receive communion.

In old Spanish colonies, the practice was by indult (Special permission) that all children or infants who were baptized were confirmed and received first communion at the same time. This was because there was a high rate of infant mortality. Since they were missionary territories, the Spanish colonies, (Central and South America, and the Phillipines, as well as a few smaller holdings) were granted this permission.

In the case of the Eastern Catholics, the priest is the ordinary minister of confirmation. In the case of Latin Rite Catholics the priest may confirm, with special permission (faculties) or if the person is in danger of death, may “presume” those faculties, since there may not be time to obtain them before the person dies, and the priest is fairly sure that the Bishop would grant them.