(Note - I did not read the entire site.) The type of tissue culture they are using at that site is bascially making leaf cuttings, which some plants like begonia and african violets (their example) can do without all the steps. If you did the same thing with a fig leaf, you would still have the virus since the virus is in the leaf tissue - and I doubt such leaf culture would work on figs. It is not the technique of tissue culture per se that rids a variety of disease, but rather very young, non-diseased tissue used in the culture.
The kind of tissue culture required to rid oneself of a virus is meristem culture, which is using only the very small meristem (actively growing apical tip - just a cluster of small cells) where the cells are actively dividing. For that, you would need a microscope to find and excise it, culture, and then divide. Methods vary.
Theoretically, you can get a separate plant to develop from each living non-differentiated cell since each cell has a nucleus that contains a complete set of genetic information that would be required to grow a whole plant. Because of this capability, growers now use tissue culture for making great numbers of plants from a small amounts of plant material.
In the 'old days' to get a new plant either from grafting or cuttings, you needed one node/axillary bud to grow into that plant. Today, you only need, theoretically, one cell. Many things are now propagated via tissue culture, but that is as much for propagating in great numbers as it is for rendering something disease free - though that is an added bonus if the right tissues are used.