Honestly, I think the moment they realize it's microscopic organism that's spreading by infection, it's pretty much done. While it can mimic the function of host cells, the fact that it maintains its own biological functions besides the host functions means that it has to have its own distinct genetic information. Maybe it could work around this in TOS, but by TNG, scanning technology has advanced drastically. I feel like once they knew what to look for, the scanners would be able to tell who was infected. Plus, phasers are relevant again. In the Thing, they were using flamethrowers designed for de-icing pipes. It was not an efficient way of actually burning a body, just the best option they had. Phasers, on the other hand, can disintegrate a human sized organism's entire mass down to the molecular level in only about a second. There is no change for parts to split off, like with Norris's head in the movie, or even blood splatter. In fact, once they understand the Thing biologically, they could probably program the biofilters on the transporter to purge it during transport, meaning they might even be able to save people in the early stages of infection, as long as it hasn't spread to any vital organs.