Let's say the probability of someone cloning a Velociraptor within the next 20 years is 1 in 1 trillion. Let's say that given a Velociraptor has been cloned, it has a probability of 1 in 1 million of escaping. Let's say that given a Velociraptor is on the lam, it has a 1 in 100 billion chance of breaking into your house. Let's say that if there's a Velociraptor in your house, checking for it increases your chance of survival by 1%.
Let's assume there are 1 quadrillion Bitcoin addresses created.
Under the assumption of correct implementation, checking for velociraptors in your closet is literally more rational than checking for duplicate addresses.
Now, there is some value in checking for duplicate keys as a safeguard against implementation bugs, but that's best covered by generating a billion addresses on your own and checking for duplicates within your personal set.
Not worth the effort. You could absolutely make a wallet that does this, but you'd have to either do a lookup with a trusted third party, or keep around a bloom filter of used addresses to query against (probably about a gigabyte). We're talking about "winning a lottery jackpot several times in a row" odds.
Blockchain's function is only to prevent double spend. Note that addresses are just a virtual term, in reality when you send money to an address you are setting up a "lock" (predicate) that can be unlocked only by a private key associated with public key that hashes to this address.
You could, but it's simply not necessary. I have not verified the numbers, but I think this answer gives a better idea of the scale we are talking about:
