One can't argue with that price. It's only about twice what I pay for soda water in quantity to dilute various beverages, including my favorite blended scotch -- not single malt.
1) For the dyslexics in the crowd, do not confuse chlorhexidine with hexachlorophene (
https://en.wikipedia.org/wiki/Hexachlorophene). The latter is much more toxic.
2) Anti-bacterial and anti-fungal activity do not translate directly into anti-viral activity. In vitro studies show only that there is antimicrobial activity, not whether that will help in actual application.
3) For SARS-CoV-2, activity of chlorhexidine compared to just hand washing is controversial:
Source: https://link.springer.com/article/10.1007/s10266-021-00660-x
"In conclusion, CHX [chlorhexidine] may present an interesting virucidal efficacy against HSV-1 and Influenza A viruses. However, reductions of HCoV and SARS-CoV-2 strains, when assessed in vitro, have not yet been demonstrated. In addition, rising [sic] with CHX may temporarily reduce the viral load of SARS-CoV-2 in patients with COVID-19."
4) Studies done to evaluate topical antiseptics are particularly difficult to do technically and are often controversial. Most important, they uniformly lack clinical studies showing any positive advantage. Distinguishing between antimicrobial activity per se versus physical removal (e.g., hand washing), versus residual activity (e.g., any soap versus alcohol) is extremely difficult. Trying to "sterilize" your hands is futile. That can easily be shown with "gloved-hand" experiments and cultures of what's in the gloves after removal.
My opinion is that hand washing is adequate. If you cannot wash your hands, then an antiseptic cleanser may help, but I wouldn't go out of my way to use it. Preparations with alcohol can cause dry skin with continuous use, which itself can add to the risk of infections to yourself or transmission of infectious agents to others.