They sit for the entire case thinking they are possibly a regular juror. Then at the end, they are told they are not needed and they can go home.

I think your proposed fix could get messy real easy since the decision making environment is changed. If people know they are alternates, they might behave differently.

With the same set of facts:
*As an alternate voter, you might vote guilty because you are independent of everyone else, and what they think has no influence on you. You're not a part of the consensus.

*But put together with 11 other jurors, you might choose to vote not guilty (because of peer pressure and all that, which is part of the process)