I thought that public addresses were meant to be exposed to the wild, so what harm is there in using the public key of your offline savings wallet?
A public "bitcoin address" and a "public key" are not the same thing. When you receive bitcoins at an address for the first time, that address is protected by ECDSA, SHA-256, and RIPEMD-160. The address is public, but the public key is not yet public at that time. If, in the future, a weakness is discovered any one or two of those cryptographic algorithms, your balance will still be protected by the remaining algorithm giving you time to move to a new algorithm before anyone can take your bitcoins.
The first time you send any bitcoins that have been received at that address, you broadcast the public key. At that point, the private key is no longer protected by SHA-256 or RIPEMD-160. It is
ONLY protected by ECDSA. Right now ECDSA is secure enough in most cases, so this isn't a concern, but for long term storage you'll want to consider the possibility that a weakness is discovered in ECDSA and you don't hear about it before an attacker does. If you've never sent any bitcoins that were received at the address it won't matter, since you are still protected by SHA-256 and RIPEMD-160. If you have sent those bitcoins and continued to re-use the address, then you've lost that additional protection.
If this doesn't concern or worry you, then you can go ahead and re-use your brain wallets. I just assumed that you were very concerned about security.
I am mostly going through all of this to protect my offline savings wallet. Therefore, it's imperative that it be immortal, so things like corruptible files, shaky hardware, and physical copies like paper wallets won't do it for me. I only trust myself and I am very happy to know that all of my savings can be backed up in my head. This is the single greatest trait I see in Bitcoin, in my humble opinion.
I suspect that paper will out-live you and your memory. Brain-wallets are typically one of the weaker traits of bitcoin. Most people don't chose a passphrase with enough entropy, and human beings are VERY bad at doing anything in a random way. We just aren't designed that way.
I only considered Brainwallets because I can create a passphrase that can ultimately bankrupt the Brainwallet concept if it is ever compromised
I'm not sure what that means, but the brainwallet concept is a rather weak concept for the vast maority of users.
I just don't understand why using the public address of my offline savings wallet is such a problem in terms of security.
The public address is fine. It's when you send a transaction and broadcast the public key that you've made the address weaker.
I thought that was the part that was designed to be shared
The bitcoin address? Yes.
The public key? Only when you send a transaction, and after that it's best not to re-use the address if you are concerned about security.
And if that is such a problem, aren't there ways around it?
Yes, as recomended by Satoshi and other knowledgeable people. Always use a private key that is generated from a cryptographically strong source of randomness, and never re-use a receiving address.
I have seen some very cumbersome and user-unfriendly tutorials on using proprietary software and transaction keys so that offline wallet keys never see the light of day. Is this the best security solution in tandem with memorizing one's private key??
Since I don't know which "cumbersome and user-unfriendly tutorials" you are talking about, I am unable to answer this question reliably.