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TL;DR: It couldn't be otherwise.

The first bullet in the OP is the correct answer:

accelerations that we deal with (notably ) are much greater than the accelerations due to the other motions that it is involved in?

Judging by the comments, many people have the gist of the idea - moreover, it was already mentioned in the answers quoted in the OP. However, the approach of trying to calculate all the possible accelerations - due to the Earth rotation around its axis, rotation around the Sun, motion in respect to the Galaxy - is a hard (if not impossible) way to prove it.

In fact, all these accelerations (and hence the pseudoforces appearing when treating the Earth as an inertial reference frame) must be small compared to typical accelerations that we experience on Earth (which are of the order of $g$) as a condition of stability of our little world.

Indeed, let us consider the acceleration due to the rotation of the Earth at angular speed $\omega$. Assuming for simplicity that we are at the equator, the condition that we can neglect the non-inertial effects is $$a=\omega^2R\ll g.$$ It is significant that characteristic everyday accelerations are of the order of $g$ or smaller, since the condition above becomes the condition that we deal with velocities smaller than the escape velocity: $$\omega^2R=\frac{v^2}{R}\ll \frac{GM}{R^2}.$$ That is, if the fictitious forces in question were comparable to the accelerations that we deal with, and orovided that these fictitious forces are due to the motion in the gravity field, our environment would not hold together.

Acknowledgement: I thank @rob for bringing my attention to this simple fact.

It is necessary to distinguish two types of motions that the Earth is involved in:

1. The rotation about its own axis
2. The motion of the Earth as a whole around the Sun, the center of the Galaxy, etc.

In fact, the OP is ambiguous in whether it means the reference frame associated to the Earth surface or to its center - in the latter case we are concerned only with the second type of motion.

Free fall
The motion of the Earth as a whole is a free fall in gravitational field. The equivalence principle of the general relativity states that in this case the reference frame can be considered inertial, because all the objects in it experience the same accelerations due to gravity, and only their relative accelerations can be detected.

Earth rotation
A reference frame attached to the Earth surface is non-inertial, and fictitious forces need to be introduced: the centrifugal force, the Coriolis force, and the Euler force. These forces can be neglected, if they are are small, as discussed in the older version of this answer (see below). Moreover, one can argue that, if these were not small, the conditions on Earth would be too unstable to allow for the existence of life.

Comment
The two points above essentially correspond to the two bullets in the OP.

Acknowledgement
I appreciate the help from all who participated in the discussion and helped me to clarify different parts of this answer.

Old version of the answer

The first bullet in the OP is the correct answer:

accelerations that we deal with (notably ) are much greater than the accelerations due to the other motions that it is involved in?

Judging by the comments, many people have the gist of the idea - moreover, it was already mentioned in the answers quoted in the OP. However, the approach of trying to calculate all the possible accelerations - due to the Earth rotation around its axis, rotation around the Sun, motion in respect to the Galaxy - is a hard (if not impossible) way to prove it.

In fact, all these accelerations (and hence the pseudoforces appearing when treating the Earth as an inertial reference frame) must be small compared to typical accelerations that we experience on Earth (which are of the order of $g$) as a condition of stability of our little world.

Indeed, let us consider the acceleration due to the rotation of the Earth at angular speed $\omega$. Assuming for simplicity that we are at the equator, the condition that we can neglect the non-inertial effects is $$a=\omega^2R\ll g.$$ It is significant that characteristic everyday accelerations are of the order of $g$ or smaller, since the condition above becomes the condition that we deal with velocities smaller than the escape velocity: $$\omega^2R=\frac{v^2}{R}\ll \frac{GM}{R^2}.$$ That is, if the fictitious forces in question were comparable to the accelerations that we deal with, and orovided that these fictitious forces are due to the motion in the gravity field, our environment would not hold together.

Acknowledgement: I thank @rob for bringing my attention to this simple fact.

TL;DR: It couldn't be otherwise.

The first bullet in the OP is the correct answer:

accelarations that we deal with (notably ) are much greater than the accelerations due to the other motions that it is involved in?

Judging by the comments, many people have the gist of the idea - moreover, it was already mentioned in the answers quoted in the OP. However, the approach of trying to calculate all the possible accelerations - due to the Earth rotation around its axis, rotation around the Sun, motion in respect to the Galaxy - is a hard (if not impossible) way to prove it.

In fact, all these accelerations (and hence the pseudoforces appearing when treating the Earth as an inertial reference frame) must be small compared to typical accelerations that we experience on Earth (which are of the order of $g$) as a condition of stability of our little world.

Indeed, let us consider the acceleration due to the rotation of the Earth at angular speed $\omega$. Assuming for simplicity that we are at the equator, the condition that we can neglect the non-inertial effects is $$a=\omega^2R\ll g.$$ It is significant that characteristic everyday accelerations are of the order of $g$ or smaller, since the condition above becomes the condition that we deal with velocities smaller than the escape velocity: $$\omega^2R=\frac{v^2}{R}\ll \frac{GM}{R^2}.$$ That is, if the fictitious forces in question were comparable to the accelerations that we deal with, and orovided that these fictitious forces are due to the motion in the gravity field, our environment would not hold together.

Acknowledgement: I thank @rob for bringing my attention to this simple fact.

TL;DR: It couldn't be otherwise.

The first bullet in the OP is the correct answer:

accelerations that we deal with (notably ) are much greater than the accelerations due to the other motions that it is involved in?

Judging by the comments, many people have the gist of the idea - moreover, it was already mentioned in the answers quoted in the OP. However, the approach of trying to calculate all the possible accelerations - due to the Earth rotation around its axis, rotation around the Sun, motion in respect to the Galaxy - is a hard (if not impossible) way to prove it.

In fact, all these accelerations (and hence the pseudoforces appearing when treating the Earth as an inertial reference frame) must be small compared to typical accelerations that we experience on Earth (which are of the order of $g$) as a condition of stability of our little world.

Indeed, let us consider the acceleration due to the rotation of the Earth at angular speed $\omega$. Assuming for simplicity that we are at the equator, the condition that we can neglect the non-inertial effects is $$a=\omega^2R\ll g.$$ It is significant that characteristic everyday accelerations are of the order of $g$ or smaller, since the condition above becomes the condition that we deal with velocities smaller than the escape velocity: $$\omega^2R=\frac{v^2}{R}\ll \frac{GM}{R^2}.$$ That is, if the fictitious forces in question were comparable to the accelerations that we deal with, and orovided that these fictitious forces are due to the motion in the gravity field, our environment would not hold together.

Acknowledgement: I thank @rob for bringing my attention to this simple fact.

The first bullet in the OP is the correct answer:

accelarations that we deal with (notably ) are much greater than the accelerations due to the other motions that it is involved in?

Judging by the comments, many people have the gist of the idea - moreover, it was already mentioned in the answers quoted in the OP. However, the approach of trying to calculate all the possible accelerations - due to the Earth rotation around its axis, rotation around the Sun, motion in respect to the Galaxy - is a hard (if not impossible) way to prove it.

In fact, all these accelerations (and hence the pseudoforces appearing when treating the Earth as an inertial reference frame) must be small compared to typical accelerations that we experience on Earth (which are of the order of $g$) as a condition of stability of our little world.

Indeed, let us consider the acceleration due to the rotation of the Earth at angular speed $\omega$. Assuming for simplicity that we are at the equator, the condition that we can neglect the non-inertial effects is $$a=\omega^2R\ll g.$$ It is significant that characteristic everyday accelerations are of the order of $g$ or smaller, since the condition above becomes the condition that we deal with velocities smaller than the escape velocity: $$\omega^2R=\frac{v^2}{R}\ll \frac{GM}{R^2}.$$ That is, if the fictitious forces in question were comparable to the accelerations that we deal with, and orovided that these fictitious forces are due to the motion in the gravity field, our environment would not hold together.

Acknowledgement: I thank @rob for bringing my attention to this simple fact.

TL;DR: It couldn't be otherwise.

The first bullet in the OP is the correct answer:

accelarations that we deal with (notably ) are much greater than the accelerations due to the other motions that it is involved in?

Judging by the comments, many people have the gist of the idea - moreover, it was already mentioned in the answers quoted in the OP. However, the approach of trying to calculate all the possible accelerations - due to the Earth rotation around its axis, rotation around the Sun, motion in respect to the Galaxy - is a hard (if not impossible) way to prove it.

In fact, all these accelerations (and hence the pseudoforces appearing when treating the Earth as an inertial reference frame) must be small compared to typical accelerations that we experience on Earth (which are of the order of $g$) as a condition of stability of our little world.

Indeed, let us consider the acceleration due to the rotation of the Earth at angular speed $\omega$. Assuming for simplicity that we are at the equator, the condition that we can neglect the non-inertial effects is $$a=\omega^2R\ll g.$$ It is significant that characteristic everyday accelerations are of the order of $g$ or smaller, since the condition above becomes the condition that we deal with velocities smaller than the escape velocity: $$\omega^2R=\frac{v^2}{R}\ll \frac{GM}{R^2}.$$ That is, if the fictitious forces in question were comparable to the accelerations that we deal with, and orovided that these fictitious forces are due to the motion in the gravity field, our environment would not hold together.

Acknowledgement: I thank @rob for bringing my attention to this simple fact.