Does boiling water really treat it to make it safer than filtered/chlorinated water?
What do you mean by "safer"? If the water is pure to start, there's no point in boiling it. For mildly contaminated water, chlorination is inexpensive, but can form some toxic halogenated carbon compounds. Boiling is very energy intensive, and may require use of fossil fuels (unless you use a solar water heater). Both eliminate most microorganisms, though there are a few, mostly nonpathogenic ones, such as Deinococcus radioduran and others found at undersea volcanic vents, that can survive radiation, heat and other chemicals. Bacteria have even been found living in the common biocide, povidone-iodine! See this source for more information on biocides.

Better alternatives might be ultraviolet sterilization or ozonation.