Why aren't Bitcoin transactions processed and verified by normal Bitcoin clients? My understanding is that the blockchain is exclusively built by miners. My question is, why isn't that verification done by every Bitcoin client - kinda like p2p file sharing programs allowed people to use the network (download) and support the network (upload)?

Update:

What is the difference between "verifying" the transaction, and the "proof of work"? My understanding of verifying transactions is that all inputs to and outputs from an address are added up, ensuring along the way that there is enough that has in-come before-hand to allow each output. If each node in the network does this, all nodes will agree and blockchain spoofers would be drowned out by the masses. Is this not sufficient proof?
Transactions are in fact verified by every network node (computer running bitcoin-qt). A transaction which is not valid will not be propagated and the client will not display it to the user as a payment.

However, a transaction being valid doesn't mean there are no contradicting transactions. What miners do is synchronize the transactions using proof of work as a signal.