When the thought arises: "I want ice cream", the "want" is fueled by craving (tanha) but the arising of the idea of "I" is attachment (upadana) and becoming (bhava) rather than craving (tanha). The Mahayana sutra you are studying says:
There are four attachments: attachment to sense pleasures, attachment to views, attachment to virtue and practice, attachment to
The Pali suttas distinguish between craving and becoming, as follows:
The craving that makes for further becoming — accompanied by passion & delight, relishing now here & now there — i.e., craving for sensual
pleasure, craving to be, craving not to be: This, friend
Visakha, is the origination of self-identification described by the
For self-identification or suffering to arise, craving alone is not enough. As described in MN 44, above, for self-identification or suffering to arise, the craving must lead to new becoming (bhava). The idea of 'I' or 'I exist' is becoming.
Here, suttas such as SN 12.12 and MN 148 clearly explain the idea of "I" arises after the arising of craving and that there can be craving without the idea of "I". Therefore, craving can be defined without mentioning self, as follows:
Venerable sir, who craves?
“Not a valid question,” the Blessed One replied. “I do not say, ‘One craves.’ Since I do not speak thus, if one should ask me,
‘Venerable sir, with what as condition does craving come to be?’ this
would be a valid question. To this the valid answer is: ‘With feeling
as condition, craving comes to be; with craving as condition,
clinging; with clinging as condition, becoming…. Such is the origin
of this whole mass of suffering.’
It follows the non-understanding of anatta is not at the root of craving; even though the realisation of anatta, together with the realisation of impermanence (anicca) & unsatisfactoriness (dukkha), will end craving.
If the mind realises the aggregates as 'worthless' (as described in SN 22.95) there will certainly be revulsion and dispassion which is nothing but the end of craving. However the realisation of the aggregates as "worthless" is not 'anatta'. It is 'unsatisfactoriness' ('dukkha'). In other words, ideas of "I" could still possibly arise even though the aggregates are declared as worthless (asāra).