"Dad" is a specific reference (when you say it you mean somebody different from when I say it), so it gets capitalized like any proper noun. On the other hand, "dad" is a common noun meaning "father" (anybody's). You only use disambiguators like "my" or "a" with common nouns ("my dad", but not "my Dad" just like you wouldn't say "my John Smith").
So the passage you quoted has it right. (3) is not capitalized because it's using a common noun ("a dad"), not a proper one ("Dad").
You have noticed a very peculiar aspect of English vocabulary. As rich as it is in comparison to many other languages, due to its almost creole history, it really is impoverished in comparison to other languages in kinship terms.
But 'why' is always a difficult question, especially when mixed with cultural questions. There are the difficulties with Sapir-Whorf type explanations: both language restricting thought on one hand and the number of Eskimo words for snow on the other.
Does the lack of kinship terms reflect the cultural lack of warmth and caring for relatives among English speakers, that is not caring leads to the loss of the words (which etymologically do exist in the ancestor languages), or did the arbitrary lack of kinship terms contribute to the crumbling of English family values?
Any direction sounds much too tendentious, too judgmental, and requires too much unjustified and biased assumptions to choose.
The comparative lack of kinship terms does ask for an explanation but one backed up by linguistic and anthropological and comparative research. The only source that comes to mind is Levi-Strauss's 'The Elementary Structures of Kinship.' (primarily anthropological but as a by product a number of examples of kinship term systems.
English isn't alone in having relatively few kinship terms. Some other European languages have only a few extra (French, German) and some languages really only have names for their clan and generation (anybody of one's biological parents' generation might be called something like 'uncle' or 'aunt', even one's birth parents).
Having no definite answer to your question, I can only say beware of making cultural inferences based on restrictions to languages. Some languages have grammatical gender and others don't, but that doesn't mean the ones without can't recognize the sex of other people.
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I can definitely personally attest to it being used by lots of working class families, middle class families, families who would be "old money" by a lot of people's standards but don't have titles, and aristocratic families where there's still ownership of the estate (as opposed to "well yeah, I'm a baroness, but it doesn't mean anything and I don't use it", though that would cover my partner, and she grew up with that use too).
There's definitely a regional element to it. My using those terms wasn't remarkable for a working class kid where I grew up, while my partner's using them did mark her as middle class where she was.
In my experience, it's the working class that have the strongest regional variation; middle class kids have a mummy throughout the UK and Ireland bar a few with a mum and a small number having a mother, while working class kids may have a mummy, but may have a ma, mam, mammy or mum. Moms seem to be more common than they were, suggesting a US influence.
The tendency to use the terms into adulthood seems to have a similar relation to both class and region; some places marking one as working class, some places as middle class, and some places giving nothing away.