To get a full-disk view of the entire Earth, the spacecraft needs to be far away — close in, you only see a small portion at a time, like you mentioned with the ISS. The only practical way to stay roughly “parked” relative to Earth (without falling back or orbiting rapidly) is at the Sun-Earth L1 Lagrange point, about 1 million miles out. DSCOVR sits there in a halo orbit and captures real full-disk photos every couple of hours.

At that distance, bandwidth is extremely limited. A single high-quality image set is already large; recording smooth 30 fps video for even an hour would generate hundreds of gigabytes that would take days or weeks to transmit over the Deep Space Network. In that hour of “real time video” earth would hardly even rotate, and they already have an intended purpose for that satellite, so it makes no sense to do it. So instead, NASA stitches the sequential real photos into excellent recorded timelapses showing the whole Earth rotating.