You would almost never use numbers in that context anyway. We all know problems are countable, but look at this:
What two problems! = Awkward
What two troubles! = Awkward
Someone might say: What a banana! or even What bananas!
What two bananas! = Awkward.
No matter what you use, that will be odd.
Mess, shame, etc. are legitimate countable nouns, according to their very dictionary definitions.
People will often say:
This is a mess.
We have two messes to clean up, one in the kitchen and one in the living room.
You’ve made a mess.
This is just another mess.
I just cleaned up one mess at work, and now I have to fix one at home.
I don’t see why people don’t think “mess” is countable. I can see “shame” because it’s rarer, but “mess” is often countable.
“Furniture” is not a countable noun. Neither is “weather.” I don’t know anything about technological (technical?) documents, but I know that with respect to English grammar, this is the rule regarding these words. In no other situation – if you can legitimately do it in those – can you refer to “a furniture.”
We refer to pieces of furniture.
“So warm a climate” makes sense.
“Such warm weather” makes sense.
“So warm a weather” is a just clumsy way of saying “a type of weather so warm” because “weather” itself, cannot be counted. It’s a condition or quality, not an instance, a single event, a unit or an object. No such thing as “two weathers.”
The idea of what is countable is that you can break it down into a single unit. This varies with some words because they can be single instances or examples of such a condition.
Illness (poor health) is common in the later stages of life. = the condition of being ill.
I had two illnesses in six months. = a bout of illness.
I cannot have one furniture or two furnitures.
*I bought two furnitures in six month. = Wrong
I bought furniture twice in six months.
I cannot have two happinesses.
I cannot have two angers.
I cannot have two clothes.
An airport, two airports.
A problem, two problems.
A thought, two thoughts.
A snowball, two snowballs.
You may sometimes see something such as this:
He had an intense anger I had never witnessed before.
What it really means is a type of anger, and it would be more correct to state that outright. You cannot have two angers, but you can have two types of it.
Even though the last answer might not sound terribly offensive, if we start adding numerals to modify “anger,” it sounds absurd.
They had three intense angers I had never witnessed before.
They had three wisdoms.
Their intelligences astounded me, and there were six of them.
In general, I think “countable” works to describe these words, even if you may occasionally encounter some that don’t sound overly offensive with an indefinite article in front. They’re not really counting that noun. If you can replace “a” with “one,” you have an argument for countability.
By the way, you can have two beauties. “A beauty” usually means a “beautiful person,” but it can also refer to animals and probably even objects.
The magazine will have two beauties on the cover this month.
2 : a beautiful person or thing; especially : a beautiful woman