Actually, I think the discussion at "Is 'have difficulties' correct" is very pertinent, but I also think I see where your concern about countability comes from.
Look at Merriam-Webster. https://www.merriam-webster.com/dictionary/difficulty
That dictionary gives five different meanings, the primary meaning being identical to the primary meaning in the dictionary you looked at. That meaning refers to a general state, "the state of being hard to do." What is general is not plural; all specific instances are included in the general, which is necessarily singular.
The second meaning in the dictionary you looked at, a "problem," is broken down into four more specific meanings in MW, but they are all types of problem, whether a disagreement, an objection, an impediment, or an embarrassment. But these specifics can be singular or plural.
My difficulty is that I have forgotten all the Latin I so painfully learned.
One difficulty is the lack of good roads; another difficulty is the lack of water, and the combination of these difficulties has reduced the district to extreme poverty.
In short, one sense of the word is necessarily singular, but the other senses of the word can be either singular or plural depending on which applies in the specific case.
A different way to think about this is that it is meanings that are countable or not countable, and, when a noun has more than one meaning, that noun may be countable in one sense but uncountable in another sense. The distinction between countable and non-countable depends on meaning.
Good question.
