Is UTXO synonymous with leftover money in an address and the only reason UTXO is even a concept is due to security concerns of not changing an address after every transaction? Hence, your "unspent outputs" must be moved to x address.

Not really. A UTXO is more like a particular bill or coin in that it's a specific "chunk" of value.

The reason UTXO is a concept is because bitcoin is designed as a set of UTXOs rather than as a balance. It's the same way cash is designed as a set of bills and coins rather than a balance. When you say, "I have $100.43 in cash", you mean that you have some set of bills and coins whose value totals to that amount. This is analogous to a set of UTXOs in bitcoin.

Or am I missing something completely?

I think what you're missing is that bitcoin was not designed to have accounts with balances as the primary implementation but to instead have a set of unspent outputs.

I read that you might have $10 balance in an address but your UTXOs might be $2 and $8 in the same address. How does this affect execution of a transaction on the blockchain?

The blockchain doesn't care about addresses. If you have two UTXOs that you have the correct key(s) to claim, you can form a transaction that claims those two UTXOs and creates some new UTXOs as its output.

Can somebody please shed some light on the difference between UTXOs and current address balance?

A "current address balance" is the total value of all the UTXOs a particular key can claim.