Everything true of a still image is true of each video frame, plus one thing that changes the problem completely: the viewer sees the frames in sequence, and the eye is far better at spotting inconsistency over time than imperfection in a single picture.
Flicker is the real enemy
Repair each frame independently and each result will be slightly different, because each frame's noise is slightly different. Individually all of them look fine. Played back, the repaired patch shimmers — a boiling rectangle that draws the eye straight to the spot you were trying to hide.
This is why a video tool cannot simply be an image tool in a loop. The repair has to be stable across time, which means treating the mark's position as something that persists rather than something rediscovered every frame.
Static marks are much easier than moving ones
Most video watermarks sit still — a bug in the corner, present for the whole clip. That is the good case: its position can be established once with high confidence by looking across many frames, and anything appearing in exactly the same pixels for thousands of frames is almost certainly the mark rather than the picture.
Marks that move, fade in and out, or animate are considerably harder, and results are correspondingly worse.
Re-encoding costs you quality
Editing any frame means the video has to be encoded again, and every lossy encode loses a little. The repair may be excellent and the output still slightly softer than the original, because it has been through the compressor one more time. Start from the highest-quality source you have — repairing an already heavily compressed clip means fighting artefacts as well as the watermark.
Expect it to be slow
A ten-second clip at 30fps is three hundred images. Doing that work locally, on your machine, is minutes rather than seconds. That is the honest trade for the file never leaving your computer.