This is about the use of well known words in semiconductor physics in the relatively new context of organic electronics. This is not always strictly justified, and at least confusing in many occurrences.
I will try not to slip in any deep debate about communities. Just let me say that obviously general physics applies both in classical semiconductors and organic semiconductors*, in particular when the output of a device - rather than its microscopic structure - is considered. It is fine to me to use whatever approach and treat these systems up bottom (semiconductors people do that) or vice versa, but keeping in mind that the constituents are covalent molecules.
Reality is in the middle, and organic semiconductors are likely to be compared to amorphous inorganic semiconductors even in the case of the firsts having rather crystal like structures. Molecules will be always therein, and a kind of hopping processes will be more or less crucial to conduction anyway.
PEDOT:PSS is in fact an ionic compound, a "poly(salt)", let me say. I want to say more than general ionomer, I do NOT want to propose a new name. It will be clear as you read. Be careful of italic.
For those not familiar with the material here is the link to it in Wikipedia: https://en.m.wikipedia.org/wiki/PEDOT:PSS
I would have placed a posive charge already in the structure, sketching a bit more than one EDOT ring.
Let's start with a neutral PEDOT chain itself. It is poly(ethylene-dioxythiophene), a conjugated polymer with quite low energy gap and low oxidation potential.
No drastic oxidative process, and even photoinduced electron transfer process (transfer to whatever available e-acceptor within the sample, purposefully added or not) lead to relatively stable radical-cations delocalised within the pi system along the polymer chain (there is an adapted or adopted name for these too, polarons).
If the anionic species formed are relatively stable too, then long living charges are formed.
In presence of very stable ionic species, either added or formed by the same process, a polimeric salt is obtained. In this polimeric salt charges of one sign are in chain, not in pending functional groups which is a much trivial case for which coincidentally polystyrene sulphonate PSS is an example.
PEDOT:PSS is right one example of the polimeric salts that I have just introduced. The double dot here means a certain ratio depending on formulation, as the PPS providing the counterions (neutrality) can be in excess. The degree of oxidation can also varies depending on formulation.
You said " As at least I understand it, the PEDOT chain is rich with holes and then is doped with PSS which has sulfonate anions that compensate these holes."
Is the use of "doped", or alternatively "then", that confuses you.
The oxidized PEDOT chain is rich with holes in the sense that is rich of positive radical-cations and they behave as the carriers. However the chain is far to be neutral, is effectively positively charged, as you know.
Actually there are positive charges (positive carriers) by virtue of the presence of the external compensating PSS counterions. The latters are strictly necessary for the so called holes to exist within the PEDOT chains.
This is not the case for inorganic semiconductors, e.g. doped silicon is inherently neutral.
PSS stabilize the charges that in its absence will be either transient or prone to further chemical reactions as for radicals are generally very reactive.
I think this description should help you in getting the subtleties of this field and those of its dictionaries.
I could have simply written that
in conjugated pi system the so called doping is a redox process, transferring electron between the system and a second material
(generally from - to respectively, with notable exceptions of which fullerenes are probably the best known examples).
The carrier in oxidized PEDOT are positive and their motion is the opposite motion of electrons residing on the highest singly occupied MO, that ideally spread all along a perfect chain. In reality along a chain their will be a distribution of segments with different effective conjugated length, each with its corresponding singly occupied orbital.
Positive charges move i.e. electrons move between these in chain or inter-chain SOMOs, which can be regarded as a band. In such a sense these carriers are called holes. They move by a deficit of electrons in the normally occupied levels.
As for the convenience of calling these carriers holes or not is up to you.
The point is that PEDIT:PSS can be regarded as a whole neutral material with mobile positive charges.
Then you said " Then of course when you apply a gate voltage, cations compensate the PSS anions instead which according to literature makes the PEDOT:PSS less conductive, but I would have thought this would make it more conductive as it liberates the holes in the PEDOT backbone. What do I seem to be missing here?".
I do not know what device structure you refer to. The conductivity of PEDOT:PSS is high enough to shorts two metals or doped Si - metal contacts, and to work as antistatic layer.
I believe that the behaviour of your diode will be dictated mostly by a second material with conductivity/mobility compatible to a transistor function.
PEDOT:PSS is a conductor tout court (between metals) or is in general applied*** as a selective contact when its SOMO levels mentioned above match (or mismatch) the relevant conduction levels of other materials.
In a sense the term semiconductor should be reserved to neutral PEDOT (although its bulk conductivity will be rather that of an insulator) while PEDOT:PSS has metal like bulk conductivity but a hole** type conduction mechanism as described above.
*it is clear what one mean saying "organic semiconductors", but I would prefer "extended pi conjugated systems". The same for "doped", I would say "oxidised (or reduced)". And so on. And not, there are not dampling bonds in molecules, even when it happens they are organic semiconductors.
**this should clarify that sometimes an adopted nomenclature is the first solution at hand ;)
***PEDOT:PSS is also reported as smoothing, adhesion promoter, etc all uses not pertinent to this Q.
EDIT: I realised you refer specifically to electrochemical transistors. Well, the above is still answering. Indeed, upon negative bias electrolyte cations infiltrate the material thereby neutralising the PSS anions. For the above discussion this reduces back the PEDOT, diminishing the available amount of carriers and as such the drain current.