In both of the bolded cases, the noun does not refer to particular cases of the things mentioned, but to the classes of those objects in general. This is the OED's second sense of "the":
II. Referring to a term used generically or universally. With a singular n[oun]....
19b. Generally, with the name of anything used as the type of its class; e.g. with the names of musical instruments, tools, etc.
This usage is commonly seen with musical instruments, where the whole class of instruments is referred to.
Bill plays the clarinet.
Could mean that Bill is playing a particular clarinet right now, but more commonly will mean that Bill has the ability to play the class of instruments generally known as "clarinet".
Bill is playing a clarinet.
This definitely means that Bill is in the act of playing a singular clarinet. (Not a particular clarinet that has otherwise been identified, but any given one.)
There were five different instruments on the table. Bill grabbed the clarinet.
In this case, a particular clarinet is being referred to (and its identity has been previously established in context, as one of the five instruments). You could think of the word "clarinet" as having a different meaning in this sentence than in the first: here meaning an instrument of the class, and in the first sentence meaning the class itself.
To use "a beach" in this case would not be incorrect, but I think would be less common in a context like this; the different sense would be the same as the difference between my first two clarinet examples. In this case, the listener (or at least this native AmE listener) would get a vague sense that the speaker is being unnecessarily specific: the context is in broad general terms about "most vacations" "in years gone by", so it makes sense to refer to "the beach" in the sense of "the class of objects known as the beach". "A beach" would almost give the sense that all those people on all those vacations had been going to a particular beach (but from context it would be clear that that was not the case, so not really an error).
On the other hand, I think one would never use "a family" in this case. I think the reason is that everyone has one family (more or less), so it would seem strange to make a non-specific reference: it would give the sense that one might go with someone else's family. It would be fine here to say "one's family", though. (I don't think there's a nuance difference between the two because the class of potential families one might go the beach with is a class of one.)