FumbleFingers's comment and oerkelens' answer prove by giving counter-examples that neither "could not have it" nor "could not have had it" implies an attempt.
Here, what I'm trying to describe is in which way "could not have it" and "could not have had it" are different. I'm doing so because I find interesting the use of "could not have had it" when two references of time are not explicitly given in the context. Let's consider both options:
(1) He had a clear idea – he needed seven years of peace and quiet in order to work. In America he could not have it. So he went back to Russia.
(2) He had a clear idea – he needed seven years of peace and quiet in order to work. In America he could not have had it. So he went back to Russia.
I think the key to answer this question is not in the modal verb could, but in the use of the perfect aspect: have it vs have had it.
Whenever the perfect aspect is used in a sentence, it makes us think of two instants in time. For example, in:
I have been reading for two hours (present perfect)
the two instants in time are now and the moment I started reading.
He could not have had it (past perfect)
what are these two instants? In the quoted text, there is only one reference of time. When a second reference of time is missing, my mind defaults to the present. So when I read "he could not have had it", I understand that something could not be had in the past, but in the present it could be possible.