Generating private key/bitcoin address based on a "password" Since the private key is basically a random large number, would I be able to pick a password and have that converted into an address?

For example: pick "mysecretpwd". Have it sha256ed, obtain a large integer that is the private key from which I can easily derive the public key and the bitcoin address.

Then for spending any funds received at that address I only need to remember "mysecretpwd", no private key storage on paper, mnemonics or anything, basically the funds are stored in my brain.

What are the security/privacy implications?
What you are describing has been done before, and is known as a "brain wallet".

In earlier days, some wallets and online website did use this system. However, it has proven quite insecure, as humans are terrible at picking secure random data for the "password", and computers are great at guessing them. Many BTC have been stolen from insecure brainwallets, and at this point we have much better and safer means of easily accessing wallets, namely BIP39 seeds.