Not true. When you are “zooming” on a digital photo that is sometimes* true, but that is because all of the sensor data is already being used in the “non-zoomed” photo anyway. But with video you are just using part of the image data, so zoom in is actually providing new data up to the pixel density of the imaging sensor. For example, my sensor is 4128x3096 pixels–1920 pixels is 1/2.15th of the sensor width. Therefore everything up to 2.15x zoom in a video is real optical zoom.
I tested that just now–I started recording, aiming at a few nearby old conch shells. I took a still frame from the photo with no zoom and one with 2x zoom. I cropped the unzoomed shot to 960x540 and resampled it to 1920x1080. I left the 2x zoom shot alone. Then I started looking around the two resulting photos for useful comparison points and surprisingly found a really interesting one on the 2x zoom shot–one part of one of the shells, the play of light and shadows looks remarkably like a profile drawing of some guy–receding hairline, shadow of a brow, nose, chin, neck, shoulder, all there. So I screencapped that and found the equivalent region on the unzoomed, cropped and resampled image. The details just aren’t there it that one–the 2x zoom image clearly has real detail that the crop/resize doesn’t . Here is a comparison between the two (the white balance changed between zoomed and not, but the video stayed in focus.) The “face” is in the bottom center of the shell.
*And it isn’t even always true with digital zoom of still photos–better cameras use built-in superresolution to get real extra data for digitally-zoomed still photos. (Most cameras with this get the multiple images by taking advantage of the slight shake of your hands no matter how steady you are trying to hold it–Hassleblad expects that you likely have the camera stabilized, so they actually move the image sensor around.)