I think because every time you address someone directly you use the vocative then 'hey Daddy' doesn't work. Since 'hey' is a form of greeting like hello, and every single time you address someone directly in a conversation you're not saying 'hi' or 'hello'.
Modern English doesn't have a vocative, so just 'Dad' or whatever is fine.
(I went vocative as well, completely forgot that there was a possibility of a possessive pronoun. D'oh!)
As "his" a causes lenition, but no change for vowels. As "her" a has no change to consonants and h is prefixed if followed by a vowel. So: a daidí = her dad, a dhaidí = his dad, a hathair = her father, a athair = his father. And just to be complete, a causes eclipsis if it means their: a ndaidí = their dad, a n-athair = their father
For words starting with consonants, lenition is used with the 3rd person possessive pronoun a to indicate "his", no change means "her" and eclipsis means "their". You can't lenite a vowel, but for words starting with a vowel, no change is used with a to indicate "his", a h-prefix indicates "her" and an n-prefix indicates "their".
a dhaidí - "his dad"
a daidí - "her dad"
a ndaidí - "their dad"
a uncail - "his uncle"
a h-uncail - "her uncle"
a n-uncail - their uncle"
Remember that the h in dhaidí isn't, strictly speaking, a h, that's just a typographical convention, and up until the mid 20th century, it was typically indicated by a dot over the consonant rather than a h, and strictly speaking, you should spell out dhaidí as dee-séimhiú-ah-eye-dee-eye-fada. The h in a h-prefix really is a h, not a séimhiú.