Validating nodes need to download and verify the signatures. This is true now, and will be true after SegWit.
{Also @GMaxwell can take it as a response as he has almost a same position }
No! It is not an exact description: after SegWit the term 'validation' gets somewhat loose defined. What Kind of validation? To what extent do you want to validate?
Being a 'full node'(Greg's term) or a 'validating node' (yours) is possible without preserving signatures after SegWit and it is not possible right now.
A full node is capable to handle temporary chain splits and to choose the right sequence by checking for double spend and not the signatures. It is what Segwit puts on the table, real working light full nodes.
Validating nodes do not need to keep signatures around after validation. This is true now (pruning), and will be true after SegWit.
{to @GMaxwell as well}
Nah! pruning is about giving up the ability to traceback the blockchain in case of critical splits and forks when one should consider long tracebacks. It is about deleting the whole transaction (fully spent addresses) and preserving (almost) just UTXO. After SegWit we have another option: ignoring signature part while preserving transactions. It is a different situation.
Anyone who has a transaction included in the chain can construct a proof that this happened (SPV proof), and this proof includes the signature that was used. This is true now, and will be true after SegWit (using the wtxid merkle tree instead of the txid merkle tree).
Well it is a proposal: Keep your proof in your pocket.
A very weak solution. Rejected.