OK

gg5, prog, meikyo

The more I look at this, the more I'm convinced it's just one sense.
Both senses in sankoku could correspond to "admonish". The "懇々と諭す" example for (sankoku's) sense 1 is also used in this GG5 example: "その店の主人は子供たちの盗みの非を懇々と諭した.　The store owner lectured children at length on the wrongs of stealing."

@@ -18,0 +19,5 @@
+<gloss>to make (someone) understand (a fault, mistake, etc.)</gloss>
+<gloss>to reason with</gloss>
+<gloss>to advise</gloss>
+<gloss>to try to persuade</gloss>
+<gloss>to exhort</gloss>
@@ -20,9 +25 @@
-<gloss>to remonstrate</gloss>
-</sense>
-<sense>
-<pos>&v5s;</pos>
-<pos>&vt;</pos>
-<gloss>to explain thoroughly</gloss>
-<gloss>to make see the reason (in)</gloss>
-<gloss>to persuade</gloss>
-<gloss>to warn</gloss>
+<gloss>to warn (against doing)</gloss>

sankoku 7e also splits along the same line.

①〔親・先生などが〕よくわかるように（やさしく）言う。「懇々と諭す」
②教えみちびく。「不心得を諭す」

Daijisen: 目下の者に物事の道理をよくわかるように話し聞かせる。納得するように教え導く。
Daijirin: 目下の者に，ことの道理を理解できるように言いきかせる。
Meikyo: 物事の道理（特に、自分の非）がよく分かるように目下の人に言い聞かせる。また、行動の指針を与えて目下の人を教え導く。
Reverso - several of the early matches use "admonish"

We do not "blindly follow the other JEs" at all. There are many entries where we have taken quite different approaches to glosses. If you want to make a case I suggest you avoid weakening it by throwing in anonymous insults.
The JEs are not alone in their translations. All the kokugo explanations are couched as ticking off subordinates to some degree. I see if you drop the sentence you quote into Google Translate it offers: "Like Shiori admonishing a girl, he speaks gently and slowly."  So poor old Googe with its massive NN analysis of parallel texts came up with the wrong translation too.
That said, there is a difference between admonishing and remonstrating on one hand, and explaining and persuading on the other. While no reference does it, I think it's probably appropriate to split it into senses.
The parallels between さとす and さとる are very loose. I know Daijirin comments "「さとる」に対する他動詞" but I think that's stating a contrast rather than an equivalence.

@@ -20,0 +21,4 @@
+</sense>
+<sense>
+<pos>&v5s;</pos>
+<pos>&vt;</pos>

I'll preface this with a question:
if this is the transitive equivalent of satoru (to understand), why must the glosses be so wildly different?

Here is an example I dug up.
Context: they are on an uninhabited island. they found an unconscious girl, brought her to a cave to safety. She just woke up and seems alarmed
「恐がらなくてもいいのよ‥‥‥別にあなたに危害を加えるつもりで、連れてきたわけじゃないわ」
詩織が少女を諭すように、優しくゆっくりとした口調で話しかける。

I'm sure there are 1000s of other examples just like that outside JE example sentences. It might even be the most common use for this word.
Now, which glosses might fit this sentence?

× to admonish　× to remonstrate　△ to explain thoroughly　△ to make see the reason (in)　○ to persuade　× to warn　○ to reason with　○ to make (someone) understand

ISTM that saito is the only JE here that has the balls to try to actually translate the japanese word. in contrast the modern JEs each trot out a similar parade of synonyms that are only tangentially related to the original word, that only fit one context.

I wish JMDICT would not blindly follow the other JEs when this happens. and boy does it happen a lot.

Food for thought, for this entry, as well as others.