Hewers of wood- the problem

In Shiur we dealt with the following Rambam from Chapter LI. The issue we raised was the analogy between the one who does Mitzvot by rote and "a hewer of wood" or a "digger of ditches".

if we perform the commandments only with our limbs, we are like those who are engaged in digging in the ground, or hewing wood in the forest, without reflecting on the nature of those acts, or by whom they are commanded, or what is their object. We must not imagine that [in this way] we attain the highest perfection; on the contrary, we are then like those in reference to whom Scripture says, "you are near in their mouth, and far from their heart" (Jer. xii. 2).

What is the meaning of this analogy? In what way is "a hewer of wood" or a "digger of ditches" a particularly apt analogy for one who does Mitzvot improperly?

To answer this question we must first note Rambam's sense of what Mitzvot are. It is this point that Rambam establishes before the analogy.Rambam characterizes Mitzvot, as a method of service or avoda, of a very particular sort:

We must bear in mind that all such religious acts as reading the Law, praying, and the performance of other precepts, serve exclusively as the means of causing us to occupy and fill our mind with the precepts of God, and free it from worldly business; for we are thus, as it were, in communication with God, and undisturbed by any other thing.

It seems that the Mitzvot are best thought of as extensions of the core elements of Avoda-Talmud Torah and Tefilla, the two sorts of Avoda that directly bring us into contact with Chochmas Hashem, or to be more precise Shechina. This point is made in the sefer hamitzvot regarding tefilla, which tellingly is defined as Avoda.

היא הציווי שנצטווינו לעבדו יתעלה. ונכפל הציווי הזה כמה פעמים ואמר

"ועבדתם את ה' אלקיכם" (שמות כג, כה);
ואמר "ואתו תעבדו" (דברים יג, ה);
ואמר "ואתו תעבד" (שם ו, יג);
ואמר "ולעבדו" (שם יא, יג).

ואף על פי שגם הציווי הזה הוא מן הציוויים הכללים - כמו שביארנו בכלל הרביעי - הרי יש בו ייחוד, כי הוא ציווי על התפילה.
ולשון ספרי:

"ולעבדו - זו תפלה".
ואמרו עוד: "ולעבדו - זה תלמוד".

ובמשנתו של ר' אליעזר בנו של ר' יוסי הגלילי אמרו:

"מנין לעיקר תפילה בתוך המצוות?
מהכא: את-ה' אלקיך תירא ואתו תעבד" (שם ו, יג).
ואמרו: "עבדהו בתורתו, עבדהו במקדשו"-

Clearly Avoda is first and foremost tefilla (Mikdasho) and talmud torah (Toraso). This point is also directly made by Rambam in the Moreh, in his choice of examples of Mitzvot in which a person fails to remain focused or "in communication with God ".

If we, however, pray with the motion of our lips, and our face toward the wall, but at the same time think of our business; if we read the Law with our tongue, whilst our heart is occupied with the building of our house, and we do not think of what we are reading;

Why then is a person who fails to remain focused in his Avoda compared to a "a hewer of wood" or a "digger of ditches"? What exactly is a "a hewer of wood" or a "digger of ditches"?

Feel free to ask questions about this "problem" part #1, so that we can proceed to the "answer" part #2, with greater clarity.