I've been playing around with Richard Kiss's Pycoin app which is clarifying how P2PK works. I see that the hash160 of the value:
(0x04) (x-coordinate) (y-coordinate) (for an uncompressed Testnet private key in this instance) gives a hash160 value used to prove ownership of the private key.
How is this detrimental for reuse of private keys if there is no issue with PRNGs providing low entropy? I understand how the Android bug was exploited (to a degree, it reused "random values"), but I fail to understand why only sharing the hash160 in a single transaction can be maliciously exploited. To clarify, I'm not talking about privacy concerns of tracking addresses through the Blockchain.
EDIT: The vulnerabilities in question (as specified in the answer are related to quantum computing and/or EDCSA weaknesses, neither of which exist)