Bitcoin's cryptographic protocol allows for leakage of information about one's Private Key, making recalculation of the private key possible?
Yes, this happens when the same key is used perform multiple signatures with the same R value - simply using the same key for multiple, sanely produced signatures with a different R value will not result in leakage, no matter how many such signatures you make.
Modern wallets use a deterministic nonce via RFC 6979.