> you could look up your own vote and the votes of your friends to confirm

You can't do that because visibility causes voting fraud.

By coercion. If votes can be checked by other people, a very large number of people (enough to change the result) will be forced under threat by someone else to "vote correctly".

The same happens if you can check your own vote, because any mechanism that lets you do that can usually be used by someone else - "give me your phone so I can check you voted for X like I told you to" (while holding a gun).

That's why free & fair elections have secret ballots, without personal identification on the ballots. To prevent coercion fraud.