Now, we are dealing still in the same subject, difference between Tathagata which is not to be found even here and now, and puthujjana who in this sense actually and in truth is to be found. While we came to agreement on that topic, it turned out to be rather superficial one, because one who understands why it is so, should also understand, that in the absence of any subject, person who is in the control of experience - Tathagata is not to be found even here and now - this refers to puggala (individual) without sakkaya, or personality, individual - “living arahat” - still functions quite well. It is so because cessation of craving is not synonymous with the ability to recognise of what is pleasent and what is painful, and acting based on such ability.
In this sense principle pain/pleasure works in the same way in arahat and puthujjana, the difference is in recognition of what is pleasent and what is painful.
Forms, sounds, odours, tastes,
Tactiles and all objects of mind—
Desirable, lovely, agreeable,
So long as it’s said: ‘They are.’
“These are considered happiness
By the world with its devas;
But where these cease,
That they consider suffering.
“The noble ones have seen as happiness
The ceasing of person.
This [view] of those who clearly see
Runs counter to the entire world.
“What others speak of as happiness,
That the noble ones say is suffering;
What others speak of as suffering,
That the noble ones know as bliss.
“Behold this Dhamma hard to comprehend:
Here the foolish are bewildered.
For those with blocked minds it is obscure,
Sheer darkness for those who do not see.
“But for the good it is disclosed,
It is light here for those who see.
The dullards unskilled in the Dhamma
Don’t understand it in its presence.
“This Dhamma isn’t easily understood
By those afflicted with lust for existence,
Who flow along in the stream of existence,
Deeply mired in Māra’s realm.
“Who else apart from the noble ones
Are able to understand this state?
When they have rightly known that state,
The taintless ones are fully quenched.”
SN 35: 37
You can try to read this, anyway perhaps I underestimated difficulty of the problem or perhaps it is knigarian who doesn’t understand something 
You say: 'But if the idea of Grasping is not applicable to the living Arahat when, for example, he is taking food,—then I am confronted with a genuine difficulty. In other words, if one cannot say that when the Arahat is taking food, he is (not) taking hold in some fashion or other, then I am faced with the difficulty of finding or comprehending what basically is the difference between life-action and other action, as of physical inanimate things’.
The first remark that must be made is that anyone who is a puthujjana ought to find himself confronted with a difficulty when he considers the Buddha’s Teaching. The reason for this is quite simply that when a puthujjana does come to understand the Buddha’s Teaching he thereby ceases to be a puthujjana. The second remark (which, however, will only displace your difficulty from one point to another, and not remove it) is that all conscious action is intentional (i.e., purposive, teleological). This is as true for the arahat as it is for the puthujjana. The puthujjana has sankhār’upādānakkhandha and the arahat has sankhārakkhandha. Sankhāra, in the context of the pañcakkhandhā, has been defined by the Buddha (in Khandha Samy. 56: iii,60) as cetanā or intention.
Intentionality as a necessary characteristic of all consciousness is well recognized by the phenomenological (or existential) school of philosophy (have a look at the article ‘Phenomenology’ in the Encyclopaedia Britannica), and though the subject is not particularly easy it presents no inherent difficulties. But in order to understand the nature of intention it is absolutely necessary to return to the notion of ‘entities’, and to consider the structure of their temporary persistence, which is ‘Invariance under Transformation’. This principle occurs in quantum mechanics and in relativity theory, and in the Suttas it makes its appearance as uppādo paññāyati; vayo paññāyati; thitassa aññathattam paññāyati, three characteristics that apply to all the pañcakkhandhā (see Khandha Samy. 37: iii,38). Intentionality is the essential difference between life-action and action of inanimate things.
But now this difficulty arises. What, precisely, is upādāna (grasping, or as I prefer, holding) if it is not synonymous with cetanā (intention)? This, and not any other, is the fundamental question raised by the Buddha’s Teaching; and it is extremely difficult to see the answer (though it can be stated without difficulty). The answer is, essentially, that all notions of subjectivity, of the existence of a subject (to whom objects are present), all notions of ‘I’ and ‘mine’, are upādāna. Can there, then, be intentional conscious action—such as eating food—without the notion ‘It is I who am acting, who am eating this food’? The answer is, Yes. The arahat intentionally eats food, but the eating is quite unaccompanied by any thought of a subject who is eating the food. For all non-arahats such thoughts (in varying degrees, of course) do arise. The arahat remains an individual (i.e. distinct from other individuals) but is no longer a person (i.e. a somebody, a self, a subject). This is not—as you might perhaps be tempted to think—a distinction without a difference. It is a genuine distinction, a very difficult distinction, but a distinction that must be made.
Desire for the aggregates leads to suffering. Craving for the aggregates leads to suffering. Describing them as burning chaff and dukkha can lead to disillusionment and the abandonment of desire and craving for the aggregates. This leads to freedom from suffering.