You can't make voting-from-home secure, someone at home could always show how they voted to a third party (allowing vote selling) or be forced to vote in a part

You can't make voting-from-home secure, someone at home could always show how they voted to a third party (allowing vote selling) or be forced to vote in a particular way (by someone standing next to them with a gun. The current vote-in-booth solution solves both of these problems. It's not the only way to promote voting, if you want everyone to vote then just make it compulsory.