If ants have an antibiotic gland, how can they spread hospital infections?
I have worked in hospitals (US) most of my life, treating both community-acquired, and more pertinently to this question, nosocomial (hospital acquired) infections, and have read many articles on the subject. 

I have never, ever seen ants mentioned anywhere.

People, flies, cockroaches and rats, yes.

Ants, no.

However, ants are vectors in a few foreign studies. The key word in these studies is potential.

As you stated,


  From the bulla, ants can groom the secretion onto the surface of their exoskeleton. This helps to prevent the growth of bacteria and fungal spores on the ants and inside their nest.


No antibiotic covers all organisms. Also, the secretions are to protect the ants against infection, not against the carrier state.

Pathogenic bacteria is cultured from ants themselves. The ant can be thought of as similar to a human: humans make antibodies to pathogens but are vectors of many diseases. Ants protect themselves via secretions, but are hosts to potential pathogens.

Part I: Review of Scientific Data Regarding Transmission of Infectious Agents in Healthcare Settings
Hospital hygiene and infection control
Ants associated with pathogenic microorganisms in brazilian
hospitals: attention to a silent vector
Ants in a hospital environment and their potential as mechanical bacterial vectors
Insect/Bacteria Association
and Nosocomial Infection