Does the absence of the son of a barren woman truly exist?

No, it does not.

One nice book on the topic that I encountered is Mario D'Amato's translation of Madhyāntavibhāga and its Vasubandhu’s commentary. In addition to providing an excellent translation, Prof. D'Amato does an awesome job retelling both texts in his own words, elaborating and clarifying the meaning. Another solid book on the topic is Mahāmudrā, The Moonlight, by Dagpo Tashi Namgyal.

Both books discuss the enlightened mind's perspective, particularly emptiness and signlessness.

In general, it is my understanding that emptiness refers to the fact that all conceptualized representations whatsoever are mere schematic models of the ontological reality and that signlessness refers broadly to the direct experience of reality beyond conceptualized representation, and specifically to direct ("intuitive") experience of emptiness thereof.

In Mahayana Buddhism there are many ways of inducing direct experience of reality beyond conceptualized representation -- most of them involve overwhelming or exhausting the discursive (conceptual) mind one way or another. The methods range from sleep deprivation, to teacher making sudden loud noise during student's meditation session, to koans, to analytical methods. The latter are particularly popular with Madhyamaka and Gelug students inclined towards intellectual speculation.

One particular pattern of demonstration involves showing the limits of basic descriptive constructs that pertain to "entity" - by deconstructing their underlying assumptions - which at that moment leaves the student with more or less clear idea of emptiness in their mind. However, in reality, the student is actually grasping onto the "sign" of emptiness (i.e. the handle of the abstraction) while mistakenly assuming to be experiencing the emptiness itself.

So the next step in the demonstration necessarily involves some sort of analytical debunking of the concept of emptiness itself, which hopefully leaves the student with no conceptual ground to stand on. The resulting experience of being "lost" is a door into signlessness.

In this passage, Tsongkhapa uses one particular technique for the second step of the demonstration, the debunking of the concept of emptiness. This particular technique employs simple logic that exploits the dependency between negation and negandum. If negandum does not exist, it says, how can its negation have any existence? The traditional example of a son of a barren woman is an example of a negandum. This impossible son metaphorically stands in for the svabhava of entities -- the intrinsic nature whose emptiness was demonstrated on step 1. This emptiness of the intrinsic nature of all phenomena (including all objects as well as the perceiving subject, "the two selves") is metaphorically compared with the non-existence of the son of a barren woman. The author says: if the son (the intrinsic nature) does not exist, how can its negation (the empty nature) not be a mere figure of speech? The purpose of this passage is not to discuss the intricacies of the notion of "existence" vs "true existence" vs "intrinsic existence" etc. - but to simply knock the concept of emptiness from under the student's feet.

Once the student lets go of the sign of emptiness and enters signlessness, he or she can hopefully see the emptiness of everything directly. This seeing involves actual direct and clear understanding of the limits of conceptual mind and the profound contrast between all models and reality. Hence in the next passage Tsongkhapa comments on the resulting state of "nonconceptual sublime wisdom", "developing profound certainty, that there is not even a particle of true existence in any thing whatsover", "eliminating the conception of true existence", "knowledge of the absence of true existence" and "knowledge of the lack of self".

According to Je Tsonkhapa, the entry to the Path of Preparation is a union of samata and vipassana on emptiness, which is also an inferential cognizer realizing emptiness. This knower if the first wisdom arisen from analytical meditation.

Since an inferential cognizer is a conceptual consciousness, this wisdom realizing emptiness realizes its object by way of a mental image appearing to it. Since the mental image of emptiness is a conventional truth, it appears to the inferential cognizer accompanied with the appearance of true existence. Thus, if one conceives, at this stage, that emptiness exists as it appears, he conceives that emptiness is truly existent in the sense of inherent existence.

Je Tsongkhapa simply means that emptiness (a complete negation) is empty of inherent existence, just as the basis of emptiness is empty of inherent existence.

The first direct realization of emptiness (a union of calm abiding and special insight as well) is the entry to the path of seeing, and it realizes emptiness in a non-dualistic manner. So it is free from both: the appearance of true existence, and the conception of true existence. But Je Tsongkhapa spoke in relation the inferential cognizer in this context.

Additionally, in the Middle-Length Lam Rim, Je Tsonkhapa posits that the 'a inherently existent mine' is utterly non-existent, like the son of a barren woman.

At that time, inherently established mine will not be found by the reasoning analyzing suchness, just as, for example, when the son of a barren woman is not observed, the mine of the son of a barren woman—eyes and so forth—are not observed.

In this context, he uses the analogy of the son of a barren woman to explain that phenomena (as opposed to persons) also are empty.