Okay, I understand much better now. Thank you for explaining.
I stumble over infinitive constructions following the uncountable forms because these constructions, to my ear, specify and instantiate these abstract mass nouns. I have responsibility and to do something clash in my mind because the first part implies "in many ways", while the second says "in one way".
Yes, I see exactly what you mean now. I think you're right. I have the same problem there, for the same reason you give.
All I can say is that when you use the web to find examples of use, you're always going to find examples that don't make perfect sense, or that sound unnatural. This is why I always stress how important it is to find contexts that reveal rather than obscure, that simplify rather than complicate.
My practical advice to you then is to ignore those confusing examples of
responsibility to do something. As you say, it makes more sense to say
a responsibility to do something, for the reason you give.
Another complicating factor here is when nouns come as part of 'semi-fixed' preposition phrases, such as
under obligation. In these cases, it is more likely that a speaker will use the uncountable noun even if he's thinking 'countably', as it were. This has more to do with lexis than with meaning, in my opinion. Sometimes we just say what sounds more natural for no other reason than just that. I don't think you need let this worry you. My practical advice to you is to follow the basic principles of meaning that we've discussed in this thread: say
under an obligation if you're thinking of it "in one way" and
under obligation if you're thinking "in many ways". And if you're stuck in between, use the uncountable version.