Though even then, the sun would never go below the horizon, it would fly off in a stright line and disappear. Going below the horizon only works on a globe
"it doesn't need to go over the horizon to disappear in the distance", but the problem is that it does go over the horizon rather than disappearing, as anyone who's seen a sunset can attest
so most visible stars are about 1000 light years away. And they're still visible. And our sun obviously cannot travel at least over 1000 light years twice a day.