You have not stated where you are applying to. I can imagine that practices vary by geography.
However, in 25+ years of industry, academic, and related work, primarily in North America, I have found that it is exceedingly rare for hiring or admissions committees to reach out to providers of written references (this is in contrast to a habit in industry of asking applicants for the contact information of references to speak with by phone instead of asking for written references).
I recall only 3 cases where I have been a bystander or participant to such a reachout. In two cases it was to confirm the reference provided was valid in the first place, and in one case to factually confirm a presumed typo that made one sentence in the reference letter have an anomalously negative tone.
Normal admissions or even jobs committees do not have the time to seek a supplementary "deep and meaningful" discussion about applicants. In addition, there would likely be concerns about procedural fairness in who and how would lead such a discussion, who would participate, and the possibility of bias if one person was reporting the outcome of a discussion to the committee. (As someone who has also hired in the private sector, I think those concerns are overblown, but so be it.)
Finally, it's not applicable in your situation I presume, but I would add that of course for senior, prestigious posts (deans, named professorships, etc.) there is sometimes back-channel discussion with people familiar with a candidate who may or may not be providers of formal reference letters, but that is in the context of a committee forming its own dossier about 1-2 leading candidates, a different story.