That’s right; they could basically sponsor a substitute.
After the Mexican-American War, the US government realized they couldn’t staff Irish units with only Protestant officers; their disgust at the destruction of Catholic church buildings during that campaign, combined with miserable conditions in general (more troops died from sickness than combat), led to high desertion rates and even the formation of a Mexican army unit composed of Irish deserters/prisoners (the San Patricios).
Thank you for the history lesson.