is it at all possible to incorporate a way for the Bitcoin P2P network to record everyone's personal wallet balances so backup, encryption, and theft are not a concern?
like, can you have a wallet address that is checked by the network and can confirm/verify its balance?
There's an easy way for you or anyone else who wants to to do it.
1. Get a paper bitcoin wallet from somewhere trusted like Casascius
2. Send some bitcoins to one of the public addresses shown on the sheet of paper.
3. Done deal. Send more bitcoins there if you want. The balance is available to anyone at any time via Block Explorer.
4. To *spend* from that address, import the private key into your Bitcoin wallet starting with version 0.4. Spend all the bitcoins from that address since you have now put the private key into a computer connected to the internet, which will compromise its security to some extent. Exactly how much of an extent depends on how good you are with encrypting your bitcoin wallet and safeguarding your computer from other people around you.
You would need to keep the bit of paper secure. Or obfuscate the private key in some manner as described in my signature, or elsewhere. But at least you don't have to worry about your online bitcoin wallet being vulnerable. I don't worry about my online bitcoin wallet at all. I don't even have wallet.dat encrypted — it's in plaintext in the default file wherever the client put it. But there again, I've only got .028 BTC left in it.