The life support belts you mention were included as an item in the FASA Star Trek Role Playing Game in the early 80s. They provide a limited supply of air, pressure, and retain heat somewhat, but are otherwise not as good as an environmental suit. They won't work for extended periods or for combat (they put up a noticeable glow around you without providing significant shielding) or stealth.
In other words, they're mostly useful when you don't need them for a long time. They do, however, serve better for highly detailed work and they are better for keeping a team in easy visual contact. They can also be worn in a 'sleep' state where they activate in an emergency (such as when the air vents).
Thus, they are primarily used as a safety and backup item. Most likely the belts were new and Enterprise was testing them to determine their limits and utility during the timeline of TAS.
Out of universe, they were created for TAS because it was far easier to draw a yellow line around the characters than to give them all EVA suits and still be recognizable.
I recall reading about this in one of the novels at some point, but I couldn't say which one.
As I remember, their use was discontinued due to safety reasons. If they failed, the wearer could be killed very quickly. EV suits were considered safer.
I don't believe there ever was a canon explanation though.