It’s the specific sense “listen intently” that was borrowed by the medical profession sometime in the 16th century (TLFi). At the time, doctors had a notorious tendency to heavily use (one might say abuse) Greek and Latin terms. This was heavily mocked, most famously by Molière (Le Malade Imaginaire), but also by Shakespeare (e.g. Caius in The Merry Wives of Windsor). Traces of it remain in various aspect of medical French, e.g. “anamnèse” (“medical history”).

“Terme de chancellerie romaine. Écouter et accueillir” you found in Littré is misleading (I won’t say wrong but the least one can say is that it is taking (very short) short-cuts).

The verb auscultare in Latin meant “to listen secretly, to spy” (Gaffiot and online Latin dictionary) and “to obey” (which is only a derivative meaning since in order “to obey” one has first “to listen carefully” to what they are told).

In Late Latin it came to mean “to compare”¹ » (Dictionnaire historique de la langue française) and that was the only meaning that remained in Old French (see Godefroy).

It was only in the 19th century that the word was used by the medical profession. TLF:

1^re attest. dans la lexicogr. : 1832 (F. Raymond, Dict. gén. de la lang. fr., Paris, Pitois-Levrault et Cie, t. 1), mais prob. empl. auparavant par Laennec (v. auscultation) qui, cependant, ne l’emploie pas dans son Traité de l’auscultation médiate; 1856 fig., supra ex. 4. 1 empr. au lat. médiév. auscultare au sens jur. « comparer, confronter » (1270, Chartae Carinthiae, V, 35 ds Mittellat. W. s.v., 1267, 66); 2 empr. au lat. class. auscultare « prêter l’oreille, écouter attentivement » [d’où la spécialisation méd. du terme] (Plaute, Amph., 1006 ds TLL s.v., 1534, 66).

The DHELL (Dictionnaire Historique et Encyclopédie Linguistique du Latin) interestingly compares the evolution of the latin words auscultare and audire (which gave ouïr in French).

Lat. auscultare, comme terme exprimant l’« attention auditive² », occupe la position ‘non résultative’ par rapport à audire, qui exprime la « perception effective ». Or, au cours de la latinité, auscultare ne cesse de progresser vers cette dernière valeur*, jusqu’à régir des compléments faisant référence non seulement à la source d’émission (auscultare aliquem qui dicit), mais aussi au message lui-même (auscultare quod dicit). Ce déplacement s’accélère dans les langues romanes, spécialement en français, à cause de la disparition de ouïr, dont la faible expression, sujette à des homonymies, demandait le remplacement.

Ausculter can also be used nowadays in a figurative sense to mean “to examine carefully”, so not only used for a close examination of the human body.

¹ How it evolved to that meaning could be a question for Latin. Although I presume that in order to compare anything you have to examine them very carefully (even if it can be by looking at it as well as by listening to it).
² My emphasis.