In addition
for LFTRs if they do overheat, a plug (imagine some ice plugging up your sink drain) melts out and the core (which is normally liquid, making the term "meltdown" meaningless) just flows out into containers which make is more dispersed, taking it non-critical (a normally operating nuclear reactor is "critical" meaning there is enough density if fissile material to sustain a chain reaction).
The reason we don't use them they don't generate enough nuclear waste. Because I am fully aware of how "tinfoil-hatter" that sounds, let me explain. In the 1960's, when much of the choices were being made about what should be researched in the field, there was a little cold-war with the Soviets going on (you might have heard about it). A core part of this was the creation of atomic weapons. One of the common ingredients (Pu-239) in a atomic weapon just so happens be in the waste products of common Pressurized Water Reactor designs.
This combined with a underestimation of the amount of U-235 was available lead to a concentration of development designs based on the dirtier, less-safe, and less readily available (0.74% of U is U-235 vs. almost 100% of naturally occurring Th is Th-232, in addition it is estimated that there is 4x as much Th as U naturally occurring) fuel.
Most of the waste from the Thorium fuel cycle is not suitable for weapons, and the excess of U-233 which is produced is mostly fed into the core to sustain the reaction. The U-233 which is above the requirements is such a small amount that it is not considered a large proliferation risk. That said, there is normally too much Pu-240 in waste from a "conventional" civilian reactor to be used directly for weapons but centrifuges can be used to remove the contaminating isotope.
There have been a couple of LFTRs built, notably the MSRE in 1964, which went critical (remember, being critical is the normal state for a operating nuclear reactor) in 1965, and operated as a research reactor until 1969. This tested many of the systems, including (and possibly most importantly) the frozen-salt-plug safety-valve, which was used to shut down the reactor multiple times. There has been a resurgence of interest in this technology lately, I have read of programs in PRC, India, and Japan to finish the necessary components to use this technology for power generation.