There are fewer and fewer fast-breeders all the time. The U.S. has never used them commercially. Saying it's about fast-breeders, thus, is a tiny bit simplistic.
Now, it is true that Chernobyl was, indeed, a breeder reactor, but that's not why it melted down...it was an early model, an overall poor design that created feedback when the operators messed up.
Fukushima, incidentally, was also an old design, but as far as I can determine was not a fast breeder reactor. In the case of Fukushima, the issue was seismic vulnerability of the backup generators and an insufficient sea wall.
That said, you're right in overall principle. Modern nuclear reactors are much, *much* safer.
My personal hope for on planet power generation is that we'll get commercial thorium plants built. Uranium is not a renewable resource, and has to be mined (with the associated risks, although they're lower than coal). There's so much thorium on the planet that it might as well be. It's in sea water. And thorium reactors can't melt down.