Chlorhexidine doesn’t seem to have any virucidal properties that I could find.

It seems like a good disinfectant, though. It’s used to treat gingivitis, which surprised me.
You're right I didn't even realize that. I definitely use it as a general disinfectant around the house. For virucidal, hospitals use Cavi-wipes, Sani-wipes, what we tend to call purple wipes. It kills pretty much any virus, TB, even COVID. There's not much odor, you do need to use gloves for it. Not for skin contact... Though I sometimes just use my hands for a quick wipe and then wash my hands afterwards. I use these wipes religiously to disinfectant my workstation, stethoscope, phones, pens, etc... Haven't gotten sick at all during this pandemic so hopefully I'm doing something right. COVID has made me sorta a germaphobe lol...not just COVID but all the other typical viruses that started showing up again when ppl stopped wearing masks. We're now seeing influenza A at the hospital where I work. I personally haven't seen the flu since COVID started because everyone was masked up. But now that it's back, I'm considering buying purple wipes to disinfect everything around my house. Seriously purple wipes are the best. I feel so clean afterwards. On the days when I feel super germaphobic, I even purple wipe anytime my stuff touches a counter or patient.