Is there any advantage to the $a \equiv b\;\;(\mathrm{mod}\;c)$ notation?

Question

Congruences modulo equivalence classes other than those defined by division remainders are ubiquitous in contemporary mathematics. It is not uncommon for a single mathematical argument to refer to multiple equivalence relationships, using ad hoc (but quite clear) notations, such as $\equiv_0$, $\equiv_1$, $\equiv_f$, $\sim_f$, etc.

In light of this (by now well-established) practice, it seems to me a bit strange that Gauss's $a \equiv b\;\;(\mathrm{mod}\;c)$ notation remains in vogue for denoting congruence modulo remainder classes. Even mathematical-typography-and-notation virtuoso (not to mention author of $\TeX$) Donald Knuth uses it (at least in the book Concrete Mathematics that he co-authored).

I'm dumbfounded by this; as mathematical notation goes, $a \equiv b\;\;(\mathrm{mod}\;c)$ (or even its parenthesis-free variant, $a\equiv b\;\;\mathrm{mod}\;c$) strikes me as singularly cumbersome. It makes me wonder: am I missing something?

Putting aside "extrinsic"¹ considerations such as "conformance to convention", "deference to tradition", "reverence for Gauss", etc., does the $a \equiv b\;\;(\!\!\!\mod{c})$ notation offer any significant benefit over simply $a \equiv_c b$?

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_{¹ This is no comment on the merits of these extrinsic reasons. They are simply not of interest to me in this post. My aim here is to focus only on the question intrinsic utility as mathematical notation, and especially as typeset mathematical notation.}

Answer 1

A few reasons immediately spring to mind:

$(1)\ \, $ The scope of $\!\pmod{\!m}\,$ often encloses more than a single congruence, e.g. including an entire line of congruences $\,a\equiv b\equiv \cdots \equiv c.\,$ Shifting this default context off to the sidelines (start/end of line) removes redundant notation that may obfuscate the essence of the matter.

$(2)\ \, $ Often times the "modulus" is a large expression, so it will not fit comfortably in a subscript on the congruence symbols, e.g. in a post yesterday I worked $\!\pmod{\! x^2+x-3}$eui

$(3)\ \, $ In more general rings one often works modulo multiple elements (i.e. nonprincipal ideals), $ $ e.g. $\pmod{n,\ a+b\sqrt{d}},\, $ or $\pmod{n,f(x)}\ $ or $ \pmod{x,\, y^2-x},\,$ which, again, are too big for subscripts, and better absorbed into (global) ambient context notation.

But if one prefers subscripts then one can introduce short names for lengthy congruences or ideals and then use these short names in subscripts.