Photoshop can do some basic editing. I do not do video editing my guess is what you want to do is not basic video editing. I do not video editing so I could be wrong. The healing brush is not a filter so can try to make the video layer a smart object and apply a smart filter and the healing brush can not be used on a smart object layer you can not change their pixels. I would think that you would need a video editing application like after effects or premier.
In Photoshop about all I can think you may be able to try is to apply a filter and or and adjustment layer, You would confine the filter and or adjustment to spots ares. Convert the video layer to a smart object layers, Add a smart filter fill the Filter mask with black. Then paint the spots position in the mask white like a feathered white dot.. Add an adjustment layer and do the same the the adjustment layer's layer mask.
If the spots moved then I feel you would need a video application.
JJ, it is a wee bit ironic, but I don't think you can do this with Premiere Pro, but it works like a charm with Photoshop. I guess Photoshop had the healing brush in the first place, and the way Photoshop handles video — the Smart Object approach — might have helped. Like you say, it is only useful if the camera doesn't move. Effects in PremPro and After Effects do track, but I wonder if that functionality is what makes it difficult to do a healing brush edit in those applications?
I do not know how the software works. In a video file most frames are not full frames they are more or areas that have change from the previous frame. When the video is played only change area are on the display. This cuts down on the amount of data that need to be recorded in a video file and drastically reduces the data rate required to play a video. If ever frame were a full frame playing a video on a network would not be possible at 30 FPS and 4k, 5k and 8k data requires 4, 5 and 8 times the data because of their frames size. There are some full frames for is in not uncommon on a network because of network traffic the there are hiccups in the network data rate and data does not arrive in time so frames are displayed without the area changed the video start to brake up. If this happens too much. Full frames recover and video playback look good. You eyes and mind also helps there your ears not as well audio suffers as well and so video and audio are streamed separately. However in a video editor the video is not encoded yet there are full frames and the framed will be also be buffered so the editor has quick access and can work in frames. Play rate and can be slowed and audio muted. during editing the renders the updated frames into the frame buffers. When the rendering is done the video files can be encoded. So modifying and encoding video can take some time. it does not need to be done in real time. Doing it in real time requites quite a machine and software that is up to the task. A Video layer is not a normal Photoshop layers and I believe that converting them to a smart object layer helps Photoshop to work on frame as though the were Photoshop layers. In effect make frame layers that acts as a single Photoshop layer. Adding a filter an adjustment, an overlay is applied to all frames in the object. Full frames can be rendered with modifications. The video can then be exported and encoded as a video file.
IMO it would be quite a feat for Adobe to extent Photoshop's Content Aware to include between frames in Video smart objects. Photoshop would be doing more than basic video editing.
I wounder if you can and if you can what stack mode blending would do to a video object? Yes just tried you can destroy your video with stack mode blending both image and audio.