It's possible for you to easily and reliably prove to a third party how you voted, by doing these things not in your kitchen but in a place and time of their choosing with them watching you put the vote in a tamper resistant envelope that they can collect and put in a mailbox. This enables buying or coercing votes, which has been a big issue historically. Ballot secrecy is a must-have feature - it's appropriate to risk it if (and only if) it's necessary to enable voting at all - e.g. the current practice in many countries, where you can vote in absentia if it's not possible for you to vote 'properly', but it's an exception forming a relatively tiny proportion of votes.

A goal there is to have absolute plausible deniability - that even if you absolutely wanted to show to someone that you voted for A, then we need to ensure that you could do that show even while actually voting for B. If you want to fill out a paper ballot and put it in a tamper evident envelope, then it's important to ensure that nobody can watch (current solutions include in-person voting, and for people who physically can't come, my local elections include an election committee visiting these pre-arranged houses and collecting ballots while ensuring secrecy, i.e. that it can't be filled by a family member). If you want to take a picture of a filled ballot, then we need to ensure that either you can very easily take a picture of a ballot filled differently than you actually voted, or that you can't take such a picture. In systems where you have multiple ballot sheets (e.g. one for each party) and put the one you choose in the ballot box, and the others (for whom you didn't vote) stay outside, then you need to ensure that either everyone can easily and anonymously leave with a set of ballots as if they voted for someone else, or either noone can leave with these ballots.

This is a problem with digital voting. For example, there are various schemes of digital voting you can verify that your vote was counted properly while still remaining anonymous without your cooperation. But it needs to ensure that the vote remains anonymous even if your cooperation can be forced or bought.