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In Star Trek, ships can communicate visually between them, video communication between captains is quite common, even from a great distance.
On the other hand, a mobile visual transmission device (remote camera to the ship) did not seem to exist. It did not appear possible for the captain to see what is happening on the planet or nearby ship.
They tried something close with La Forge's VISOR, but it never went anywhere.
Why was live streamed video technology absent?
Out of Universe the answer is probably a combination of Science Marches On and Storytelling aesthetics.
In the original Star Trek it is more than possible that simply no one considered the possibility of a hand held video phone. The bridge screen ship to ship communication was pretty futuristic seeming at the time.
Later on in Trek development, while hand held visual comms should have been within easy intellectual reach, the creators may have felt constrained by the original series aesthetic that restricted video communication from being "mobile".
Next up on the list of problems would be video phones are cumbersome and intrusive. Having a character holding a video comm restricts how you can film it, you need to be able to see the actor holding the comm, and to keep post-production costs down, unable to see the person on the other end. And frankly it looks ugly to have someone holding a phone out all the time. Having an unsourced video (floating nanobot cameras) certainly should have been possible particularly given the level of tech seen around Borg Nano probes, but can cause interruption of Suspension of Disbelief, when people ask where the camera is.
Finally from a dramatic story telling point of view, being unable to see the people in the away team can provide dramatic tension for the bridge crew and audience which would be lacking if they were all yelling "Behind you!" as the away team is ambushed, trapped by a rock slide or simply interrupted by another conversation.
