Yes, TB can affect bones (in extrapulmonary or more precisely osteoarticular TB), but that form is relatively rare nowadays [even more so in the West], and even rarer for the sternum to be affected. On the other hand, it is one of the ways in which ancient skeletons (or mummies) were diagnosed with TB, but sometimes confirming by some other means (e.g. PCR), as noted in a 2013 review
Bone and joint tuberculosis (TB) is an ancient disease. Evidence of osteoarticular TB has been detected in Egyptian mummies [7, 22, 70], Iron Age remains from Asia [63, 64], and skeletons of Europeans living in the Middle Ages  by histological or polymerase chain reaction (PCR) study [70, 71]. [...]
Bone and joint TB currently accounts for 2.2–4.7 % of all TB cases in Europe and the USA and around 10–15 % of EPTB [extrapulmonary tuberculosis] cases [...], whereas in undeveloped countries, particularly Asia, the incidence of EPTB increases to 15–20 % .
TB is one of the most common causes of rib osteomyelitis. [...]
Tuberculosis of the sternum is rarely reported and accounts for 1.5 % of bone and joint TB cases.
And from a more [paleo]history oriented paper:
Paleopathological diagnosis of tuberculosis (TB) essentially relies on the identification of macroscopic lesions in the skeleton that can be related to different manifestations of TB.
Alas the rest of the paper discusses tuberculous meningitis and its identification from skull impressions.
Another paper on diagnosing ancient skeletons with TB notes that a criterion is:
lesions that are mainly destructive, producing cavitation
into the cancellous bone
and on prevalence in older times:
data from the preantibiotic era indicate that about 5–7% of cases showed bone
changes (Steinbock, 1976)
As far naming goes, according to Wikipedia, TB
was historically called consumption due to the weight loss.
So that popular name was not due to effects on bones.