I can't find the CDC page you allude to, but I suppose you've read something like this:
Cleaning, accomplished with soap–or detergent–and water, refers to the physical removal of dirt and grime, and in the process, some portion of the germs on a given surface. Sometimes cleaning tools, including sponges and cloths, simply move germs from one surface to another. Disinfecting, on the other hand, refers to killing a high percentage of the germs on a surface or rendering them incapable of reproducing.
Sanitizing is another relevant term in this discussion. [...] sanitizing lowers the number of germs on surfaces to a safe level, as judged by public health standards or requirements. The process works by either cleaning or disinfection to reduce the risk of spreading infection. Finally, sterilizing destroys all forms of microbial life and is used mainly in healthcare and laboratory settings. [...]
If disinfecting a surface is the task at hand—for example, you might be tackling a food preparation counter that has just contacted raw meat or fish—the order of operations is important. Just remember “C” (cleaning) comes before “D” (disinfection).
The point is largely that if you do it "the other way around" you risk actually [re-]contaminating the surfaces, e.g. with a sponge that isn't so clean. And as Bryan Krause says, a heavy buildup might not even be penetrated sufficiently in the decontamination step.
Depending on the target virus, that page recommended various times for leaving the disinfectant (bleach) on the surface, e.g. flu for 10 minutes at the concentration they suggested, and for norovirus "until dry". Also, after bleach disinfection
If object is intended for food or mouth contact, rinse with plain water before using.
This is because bleach leaves a solid residue that may still contain active substance(s).
Having said that, it's not impossible to disinfect your hands, but you have to use a product that's suitable for skin use, e.g.:
- "iodine" (iodopovidone)
- hydrogen peroxide
- alcohols (at high enough concentration), although this source says they only have "minor activity against some viruses".
And some more listed there... with a discussion of the most suitable pathogens; the ones I've picked above were mentioned as reasonably effective against viruses. (Some of these give burning sensation if you have any cuts or scrapes.)
Regarding disinfection of hands, the CDC says that in general
Many studies show that hand sanitizers work well in clinical settings like hospitals, where hands come into contact with germs but generally are not heavily soiled or greasy. Some data also show that hand sanitizers may work well against certain types of germs on slightly soiled hands. However, hands may become very greasy or soiled in community settings, such as after people handle food, play sports, work in the garden, or go camping or fishing. When hands are heavily soiled or greasy, hand sanitizers may not work well. Handwashing with soap and water is recommended in such circumstances.
Interestingly, that CDC page is heavily in favor of alcohol(s) as a general hand disinfectant, although it only talks about "germs" in general, without getting into sub-categories like viruses, bacteria etc.