The reason your identity becomes exposed by spending from a wallet is because the IP address (internet, not wallet address) where you transmit the UXTO to the mempool from is logged by third parties. They say you shouldn't reuse a wallet address, but they mean for receiving funds. You shouldn't give out the same wallet address to multiple people, because then any of those other people that send a sum to that address will know who you are, and they can see addresses you've received from, spent to, and your balance.

You cannot brute force a private key if you have a public key. This is the entire theory behind public key cryptography, that given a sufficiently large keyspace (bits of encryption) you can give out your public key to anyone in the universe, and they can never ever figure out your private key even if they converted every atom in the known universe to a transistor and dedicated this amount of computing power for trillions of years to finding your private key. This was true for several decades until some smartass mathematician figured out that QC actually has a 'backdoor' into finding your private key, don't ask me how because its all quantum magic. No wonder Nobel didn't like mathematicians.

The public key to your wallet actually MUST be shared and stored publicly on the blockchain because when each full bitcoin node scours (verifies) the blockchain, it also checks the balance of each particular wallet address. It does this by using the public key for that transaction to check the signature block. When a signature block is created by the owner of the wallet, they use their private key to sign the transaction which contains the amount of bitcoin to transfer and to which wallet address.

If the public key for every wallet address weren't immediately available to each and every bitcoin node, there would be no way to verify that a transaction was authorized by the wallet owner. The shorthand that was chosen by Satoshi was to use the actual ECDSA public key encoded in BASE58 as the wallet address itself.

Base 58 is a standard agreed-upon way to encode a binary number as a string of characters and numbers where some characters like O and I aren't allowed because it can be confused with 0 and 1. If a bitcoin address used all uppercase, lowercase, and digits it would be Base62 (26+26+10 = 62) or base64

The reason ECDSA (Elliptical Curve Digital Signature Algorithm) was chosen over RSA is EC has a much smaller public and private key size and you can use a much shorter string of characters (BASE58) to convey the public key for the ledger entry.

I hope I didn't lose you in all of this.