One of the properties of German nouns is 'Zählbarkeit' (countability). Traditionally, German grammars (and some dictionaries) make a difference between countable and uncountable nouns and take it as a fixed feature, e.g. 'Obst' being uncountable and 'Frucht' countable.
Christian's 'Rettich'-Example is a good demonstration of the fact that a noun's countability can be influenced and even 'abrogated'. Normally, 'Rettiche' can be counted, but you can also use it as an uncountable mass noun like in
ich mag Rettich nicht
You could also say
ich mag Banane nicht
As a result I think that (un)countablity is more an aspect than an unalterable property of a noun. Guidot calls it 'regarding a noun as abstract'.
Ich finde, Brille steht dir nicht.
The sentence doesn't mean the specific glasses the person is wearing now but any
glasses at any time, glasses at all. It works like an adjective, like 'Rot steht dir nicht.'
This example may be interpreted as a generalisation, which is also the case for 'mit Karte zahlen'. One could say 'mit meiner Karte zahlen', with one specific individual of a cash card, but that is not what he wants to say, he just doesn't want to pay cash.
Another point ist that there are numerous examples for the integration of a noun into the predicate by omitting the article. In those cases a noun is not regarded as a noun in the proper sense but as a part of the verb like a separable verb particle:
'ich fahre nicht mehr oft Auto, das ist mir zu stressig'
'sie fährt jedes Wochenende weg'
Before the spelling reform some of those predicates had to be written in one word like 'radfahren' and 'maschineschreiben'.
In the same way, there are many 'Funktionsvergefüge' without an article like 'in Frage stellen' or 'Kritik üben'. So 'mit Karte zahlen' could be a 'big verb'.
Finally, the prepositions 'mit' and 'ohne' might tend to be used without an article like in
versuch's mal mit Schraubenzieher,
which is not standard German but everyday spoken German. In the same way you can ask
fährst du nach Karte oder mit Navi?
In addition to that, you can find more or less idiomatic uses of the 'Nullartikel' like 'vor Arbeitsantritt', 'wegen Trauerfall geschlossen', 'bei Stau fahren Sie am besten ...', but they appear to be generalisations.
So, 'mit ø Karte zahlen' may be explained
- as a generalising 'Nullartikel'
- as an abstraction from the specific, concrete bank card involved in the payment and at the same time as a suspension of countability
- as a sort of adjective: 'nicht bar, sondern mit Karte'
- as an integration of the noun into the predicate
- as a specific potency of some prepositions like 'mit'
- as a half-idiomatic way of constructing a 'Modalergänzung'