Especially on first contact missions they should have someone from the Enterprise filming, recording for legal and historical reasons. Perhaps yeomen would do this and "historians" would catalogue the information.
To me they should have yeomen recording every mission but perhaps they do subtly like in "Wink of an Eye" where they had recordings of the crewman's disappearance.
That was theoretically what yeomen were supposed to do, which was why they carried tricorders. A yeoman is basically the captain's secretary (the naval equivalent of the company clerk job that Radar and Klinger did on M*A*S*H), so their job on an away mission would be to maintain an audiovisual log of the entire thing. Unfortunately, that idea got lost in the shuffle along the line.