When analyzing sentences for meaning, it's necessary to use common sense as well as the niceties of punctuation and syntax.
It should be clear that in the first sentence, the nonrestrictive relative pronoun which refers to the entire preceding clause (or perhaps only to A good watchdog barks loudly when strangers come) and not to property: that makes no sense in context. Perhaps it would be clearer if the sentence were:
A good watchdog barks loudly at strangers who come onto your property, which gives you a feeling of security.
This sentence can more easily be broken into meaningful parts:
(i) A good watchdog barks loudly, which gives you a feeling of security. [The prepositional phrase may or may not be necessary to give you a feeling of security, but if it's not there, it suggests that a loud bark (perhaps for no apparent reason) is what makes you feel secure.]
(ii) at strangers who come onto your property [This part defines the strangers and restricts the dog's barking only to times when they enter your property, so I think it's necessary for defining a "good watchdog" in this sentence.]
Your second sentence says exactly what you object to about the first sentence. Without the comma, the relative pronoun is restrictive and refers to "property that makes you feel secure". This is a good example of the ambiguity of that "that" and "which" and the absurdity (to borrow your judgmental term) of believing that they are always interchangeable and that it doesn't matter which one you use because readers will always be able to figure out what you mean.