Index Fungorum is not a good source for infraspecific taxa as they synonymize all of them to their type. Now it should be assessed if the variety is useful or not. I would tend to think it is useful as this variety is restricted to a geographical area and very different morphologically from the type.

Agreed, for some reason Clitocybe squamulosa var. montana was omitted from the original swap. It should be moved to Infundibulicybe with the others.

I think the cost of having a name combination like Infundibulicybe squamulosa var. montana that isn't published when its implied by the hierarchy (e.g. descending from Infundibulicybe squamulosa descending from Infundibulicybe)
is much worse than the cost of having trinomials (ssp) and binomials (sp) grafted to parents that don't have consistent names, especially when we have no reference to follow, as in the case with Fungi

The main problem we need to solve is that all ssp downstream of Infundibulicybe squamulosa should have the format Infundibulicybe squamulosa ssp. xxxxxx
Clitocybe squamulosa ssp montana should only descend from Clitocybe squamulosa

Just for the record I don't agree that invalidly published names should be added - at whatever rank.
What is the difference between allowing invalid trinomials such as Infundibulicybe squamulosa var. montana, or effectively creating dozens of invalid combinations at species rank for Aecidium in Puccinia?
https://inaturalist.nz/taxa/84769-Puccinia
Please don't do that. Plant pathologists would not be happy.

There's a thing I can't understand in this situation. Is the fact of having inconsistent trinomials a technical problem or just a policy problem? It feels weird to have to make up names for trinomials while it's strictly forbidden for higher ranks. And keeping the old name feel just as weird as literature updates to these names and this creates disparities between iNat and literature... Or is there something that I'm missing?

I am a member of the International Commission for the Taxonomy of Fungi (ICTF) and one of our groups is looking at the issue of a globally consistent checklist. However, the current view is that whilst possible for higher taxa, it is unlikely at species level. There are too many species, too few mycologists, and still too much change due to the shift from morphology to phylogenetic-based species concepts. Those issues are combined with the still on-going consequences of the abandonment of dual-nomenclature (different names for sexual/asexual forms) some years ago. So, the situation is unlikely to change and especially impacts iNat 'popular' fungal groups that aren't the focus of most pathology-based taxonomic effort (i.e., where most mycology funding goes). I did think the iNat open curation system at least demonstrated a good model for a way forward, although recently I've become less certain of that.

I do think that having, for example, all Aecidium species placed under Puccinia is a sensible way forward given the current iNat taxon data model. It is both nomenclaturally and taxonomically correct, even if it looks odd. To leave a synonymised genus present simply to provide a parent name for uncombined child taxa inevitably leads to incorrect use of that genus for observations and an invitation to double-up synonymous species names. Leaving legacy synonymised genera also doesn't provide the very helpful nudge that Aecidium is really just a different life-stage of Puccinia. The alternative of creating effectively invalid combinations is simply wrong (under ICN) and will lead to even greater confusion, especially when these fictitious names 'collide' with the GBIF/CoL hierarchy. There would be kick-back from the professional mycological community if these invalid names leak. In the iNat fungal hierarchy you will find very many examples like Aecidium/Puccinia so this isn't an isolated example. Trinomials add to that.