That doesn't preserve the anonymity requirement for safe voting systems. Blockchains are only pseudonymous and a malicious landlord, employer or family member could easily coerce you into divulging your ID.
If you take the key with you after you vote, which you will need to do if you want to validate your vote later, then you can be coerced into disclosing it.
I believe that the keys would just be taken by the machine, that way you'd be able to verify the vote was placed without the risk of revealing the identity of the voter attached to that card.