I wonder if it would be feasible to include the hash to timestamp in the input and not in the output.

Let's assume we would change scriptSig to

timestampScriptSig:

and under the conditions that
1. The so modified input is still able to claim the output

and optional these nice to have features
2. ideally also signs the
3. the output can be empty (i.e. the whole BTC input amount is the transaction fee)

This would come with the following advantages: It does not burn any bitcoins. The transaction fee is an incentive for miners to include the transaction. And finally: You only need one transaction.

Can anyone comment if it's possible that also signs (maybe with OP_CODESEPARATOR)? Is a transaction containing timestampScriptSig a valid transaction with bitcoin 0.8.3? Would it, together with , be included in the blockchain?