Usefulness of Madhyamaka to personal practice? In my opinion, the Sutta Pitaka and Mahayana Agamas (the Mahayana equivalent of the Pali suttas) discuss the nature of the self (anatta, empty of self, dependent arising) in such a way that it is useful to the path towards liberation from suffering (such as overcoming self-view or sakkāya-diṭṭhi). 

Meanwhile, the Sutta Pitaka and Mahayana Agamas (the Mahayana equivalent of the Pali suttas) do not discuss the nature of other non-sentient things (e.g. a chair, the universe, atoms, five aggregates), beyond noting that other things are conditioned (sankhara) and impermanent (anicca).

On the other hand, Nagarjuna's Madhyamaka seems to have expanded on these concepts to cover the nature of all reality (empty of inherent nature / svabhava, dependent arising, dependent designation). On top of this, Madhyamaka also posits that emptiness itself is empty of inherent nature.

To a non-Mādhyamika practitioner, Madhyamaka's discussion on the nature of all reality may appear to be an exposition on the "poisoned arrow". That is to say, it is not useful to the path towards liberation from suffering. Some might even say that the nature of all reality falls under the fourth imponderable (Acintita Sutta).

From a Mādhyamika practitioner's perspective, how is Madhyamaka useful to the path towards liberation from suffering? How is it useful to personal practice?

This should not be seen as negative criticism towards Madhyamaka. Rather, I seek to understand its usefulness to personal practice.
Madhyamika posits that both persons and phenomena (i.e. phenomena other than persons) are empty of true existence. While traditional text do not mention the emptiness of chairs, table, etc. They do say that persons and phenomena [other than persons] are empty of true existence. The Heart Sutra says:


  Phenomena also are empty.


This citation means that both persons and phenomena are empty of true existence. Thus, it is not true that "Mahayana Agamas do not discuss the nature of other non-sentient things".

In his Commentary on Aryadeva’s Four Hundred, Chandrakirti says:


  Through the division of phenomena and persons [selflessness] is understood as twofold, “a selflessness of phenomena and a selflessness of persons.”


In addition, traditional Mahayana texts explicitly posits that phenomena one encounters and "deal with" on the path to enlightenment are empty. For instance, the objects of abandonment are empty, nirvana and samsara are empty, afflictions are empty, minds are empty, and so forth. Why? This is because while our misconception of the nature of a cup is not that which binds us in samsara, seeing the afflictions, etc. the way they exist is most relevant.
phenomena the way they exist is relevant, while it is not so much our misconception of the nature of a table that binds us in samsara.

Nagarjuna’s Seventy Stanzas on Emptiness says:


  Compositional activity is caused by afflictive emotions.
   The afflictive emotions that motivate actions have a nature of [contaminated] actions.
  The body is caused by [contaminated] actions.
  All three are empty of inherent existence



The types of phenomena of which emptiness is usually emphasized, though, are the aggregates. According to Madhyamika tenets, one cannot obtain even the path of seeing without directly realizing both the emptiness of person and that of phenomena [simultaneously]. Here, phenomena refers to the aggregates. 

In Precious Garland, Nagarjuna says:


  As long as conception of the aggregates exists,
  So long is there conception of an I in them.


This is because we posit that the conception of self of person is based on the conception of self of phenomena, and that it is not possibly to eradicate one [ignorance] without eradicating the other.