The shadow of the earth, when cast on the moon during a lunar eclipse, is round. This was known in ancient times. But note that lunar eclipses do not always happen when the moon is in the same position in the sky. If the earth were flat, the earth’s shadow would not have the same shape when the moon is directly overhead as it would when the moon is closer to the horizon. Since it is possible to observe multiple full and partial lunar eclipses during an average lifetime, this would not have been lost on any observant person, even in the distant past (note that this is assuming one model of a flat earth, the one where the sun and moon cross under the earth as they travel back to the east). Also, if the sun and moon were orbiting overhead, as in some recent flat earth models, then how could the earth ever get in between them to cast a shadow in the first place?
The moon’s phases are also proof that it’s orbiting a global earth. In real life, at any given part of the moon’s cycle, all people on earth see the same phase, and the moon is always about the same size. This makes sense if it were orbiting the globe from a distance far greater than the earth’s diameter. If it and the sun were always orbiting above a flat earth, as per a recent flat earth scenario, then they would both change size drastically, and people in different areas would see different moon phases. Someone looking toward the moon would be presented with a different view than someone looking at it from the other end of the earth.
Instead, a person with a telescope can actually watch the shadow creep across the moon’s surface and someone on the other side of the earth can then pick up watching the shadow creep when the moon drops below the horizon of the first person. There are human eyeballs all over this globe of ours, and the progress of the moon (and sun) represents a continuum.
Also, by watching the shadows creep across the craters and plains of the moon, it is clear that the moon is a sphere. Also, the crescent and gibbous phases, which have curved boundaries, are possibly only on a spherical moon, not a disk.
And we can watch sunspots migrate across the face of the sun and they behave as if they are moving across a spherical surface. There is also the phenomenon of limb darkening, where the sun (and other stars) appear darker and redder towards the outside, which proves a spherical sun, not a flat disk. We have overwhelming evidence that the other bodies in the solar system are spherical. And we have overwhelming evidence that the earth is also a sphere.
Note also, if the moon and sun were flat disks, then their apparent shape would become more elliptical as they moved lower in the sky. The fact that a full moon and sun always look circular from any angle shows that they are spheres, not disks.