Personally, I hope all fungal names do not get added to iNat. Only those names required to support observations are essential. The rest represent a significant and unnecessary curatorial overhead to keep them up to date, with no practical benefit within iNat.
At the moment iNat has entres for about 10% of described fungal species (the 10% likely to be seen and identified by iNatters). In terms of published names, rather than taxa, then iNat has around 3% of the names.
For fungi if you want a complete list of names then you should get data from IndexFungorum. That is a nomenclatural database with all names ever published. There are also several taxonomic databases focussing on lichens for various regions.