I'm a fan of simplicity too; Unfortunately, your proposal above to change the semantics of OP_CHECKSIG is too naive.

The SIGHASH data used in both the original Bitcoin script and in Segwit script contains data indicating which input is being signed.  In Bitcoin script, the input is being signed is indicated by the input that has a non-empty scriptSig field.  In the Segwit script, the outpoint corresponding to the input being signed is explicitly included in the signature data. By signing only the transaction id, your proposed signature does not include the data that tells which input of the transaction is being signed.  Thus if different inputs share the same public key due to key reuse, then the signatures on those different inputs will be identical.  Your Flexible Transactions proposal opens up a new line of attack against Bitcoin that doesn't currently exist.

Consider the following simple example, suppose you and I are jointly preparing a transaction to mix our coins, or perhaps we are jointly funding some purchase.  We jointly prepare a transaction with one input from you and another input from me.  We each sign the transaction and hand the signature data over to each other so we can produce a completed transaction.  But oh no! I lied to you. I didn't use my own input to the transaction.  "My input" was actually the outpoint from one of *your* transactions; one that has the same public key as the input you have chosen.  Now I copy your signature you have provided in your input to cover "my input", which is really your coins.  Surprise, it turns out you are funding both inputs to our "jointly" funded purchase.  Other protocols are likely similarly broken by your Flexible Transactions proposal.

I personally rate this flaw as about the same caliber as the transaction malleability you are trying to fix.  Sure, with enough vigilance, perhaps you can detect and avoid this trap.  However, it requires a bunch of unexpected work.  You must always examine every other input to a transaction you are about to sign to make sure that it isn't one of your inputs, which means you probably need a copy of the UXTO set to lookup outpoints, which is a huge burden, especially if you are a hardware wallet.  If you are not vigilante, your funds may end up stolen. Surely it is better not to open this line of attack.

For the most part, the SIGHASH works the way it does in Bitcoin for a reason. You cannot simply throw away the parts you don't understand or appreciate.  You should take the time to learn why things are the way they are, and then, only once you are certain that some aspects are not, or no longer, needed then can you propose removing them.