There are quite a big list of natural products with antiviral properties.
Most of those herbal antivirals unfortunately will be poor antivirals in vivo, when you take them orally.
This is because most studies that test herbs, supplement and other compounds for antiviral effects are in vitro studies: they use a cell line infected with a virus to test antiviral potency. A lot of herbs and compounds will have good antiviral effects in vitro at certain concentrations.
However, the concentrations used in vitro are often not obtainable in vivo, when you take the drug orally. This is because even for herbs and supplements, there is a maximum safe oral dose, and if at the maximum dose you cannot achieve the concentrations in the bloodstream that were used in the cell line in vitro, then the herb or supplement is not going to have a good antiviral effect in the body.
Like you, I collected studies on lots of supplements, off-label drugs and compounds that have antiviral effects on viruses, but from some calculations I was doing recently, most unfortunately do not seem to work in vivo.