The basic issue is regulatory. This technology is completely different from everything that the NRC and nuclear engineers know. No water, no steam, no solid fuel, no high pressures,..
This argument can be used against literately any progress of any kind. There is a long list of reason which I could give you, but none of them is neither very good nor convincing, at least in the current paradigm (which mind you changed dramatically since early 1970s).
The principal reason why was this abandoned in the early 1970s was that Alwin Weinberg, then director of ORNL, started to publicly question safety and sustainability of the light water reactors. Mind you he was one of the inventors of this design, which made matters worse obviously. He got consequently kicked out of ORNL, and the molten salt program - his brainchild - was abruptly killed shorty thereafter.
ORNL was the only lab which was pursuing this, nobody else really knew much about it, and it fell into obscurity. As the concept was reviewed within GenIV nearly 3 decades later, some European groups went to ORNL and made private copies of the research papers, and resumed their own (small scale - dozens of people maximum) research within GIF.
Kirk Sorensen managed to get NASA funding to scan and PDF most of these documents, and put them on the web in late 2006. Since then the information is publicly available with ease. I would say that the fact that Chinese started a high priority $1B national effort to build these reactors on their own within few years after this information became available, speaks rather favorably about merits of the concept.
There are reasons why to use "denaturated" molten salt reactor, that is with U238 in the core. The principal one is further enhancing proliferation resistance, such as if you want to sell these reactors to potentially unsafe countries. The original DMSR is a converter with breeding ratio ~0.8, so it needs some fissile fuel to keep going. This is still much better than light water reactors with BR of 0.2-0.3.
It is possible to go to fast(er) spectrum by eliminating the graphite moderator (which is currently favored by the French research), using less moderating salt than FLiBe, and/or increasing the fraction of heavy metal (nuclear fuels) dissolved in the salt. There are limits to the last one, as the solubility of tri-fluorides (mainly PuF3) is limited, so to run U/Pu cycle in molten salt we either need to increase salt temperature to ~800C, or to use different salt. Chloride salts are excellent for fast MSRs, but much less proven than Fluoride ones.
I do not think we need to consume U238 in MSRs, at least not in the next centuries, there is plenty unused thorium, as a waste from rare earth mining, so we can let the chloride reactors as a task for future.
However, it is possible to use small amounts of transuranics - the problematic waste from current reactors - as a fuel in regular MSRs, as a means of waste disposal. Chloride salt reactors would be better at that potentially, but are not necessary.
PS: All your questions are answered in better detail in the resources linked in my first post.
Perhaps one more, a well written first hand account of the original research:
H. G. MacPherson: The Molten Salt Reactor Adventure
Check the slides and the talk by Robert Hargraves I linked above.