You can do that locally once you have downloaded and verified the blockchain. You cannot do that to the blockchain as a whole because I don't know whether the transaction "Bob->Charlie (25BTC)" is actually legitimate when I am syncing a new node. For me to check that it is legit, I need to know where Bob got the output to spend. Just because a transaction is in a block with a valid proof of work does not automatically mean that all transactions in the block are valid; that's not how Bitcoin works.
You can certainly do this locally as that is basically what Satoshi suggests in the whitepaper. But as gmaxwell pointed out above, what we do now for pruning locally is way more efficient than what Satoshi suggests. Satoshi suggests that we throw away parts of blocks as UTXOs are spent. But what we do is that we maintain a separate database with our UTXOs and chainstate data so we don't actually need to have the blocks themselves. So we just throw away old blocks entirely because we have validated them and taken the things from them that we need and stored them elsewhere in a more compact form.
Why would you need where Bob got the output? If that transaction is included in a verified block, it means that it is valid. And if that block is like 1000 blocks behind the current block, it's impossible to change it.
(I am not talking about how the current bitcoin protocol works).
I was thinking of something like transaction cut-through
https://bitcointalk.org/index.php?topic=281848.0 which is already being implemented in mimble wimble.
