I don't consider it easy to justify the distinction between demonstratives and determiners, since as far as I can tell, syntactically at least, there is none.
But your example sentences show exactly why English has all these words:
The person went to the store
It's a good sentence to use in a context where there really is one obvious candidate for "person", and one obvious candidate for "store".
This person went to that store
It's a good sentence to use in a context where there might be more than one person and/or more than one store. I'm sure you can agree that the cognitive load is (at least somewhat) higher when trying to parse this sentence.
Obviously the two sentences don't mean the same thing in current English, but I'm wondering what if they did.
the and we didn't have the word
the, then some other construction would come about to fill the gap left by
this. Something like
this here, which would probably contract and turn into something like
thisser. But this is all very speculative.
But maybe "meaning" isn't the best word to use to describe these words, since they don't refer to objects, actions or anything else. Their use is more a discourse marking tool. The distinction between
this/that/these/those is useful for giving a hint:
the person - I expect you know which one I mean
this person and
that person - I am explicitly telling you which person.
a person - I don't expect you to know which one because this is new information to you
I pointed out that there is cognitive load associated with your second sentence above. It's efficient to reduce cognitive load, isn't it? so in the commonest case,
a are used. And that gives the listener a lot of what s/he needs to go on.