The only reason I can see for splitting out "baton" into a separate entry is that [rK] on 采 only applies to the "dice" sense. If it weren't for that, a [restr=采] tag on sense 2 would have sufficed.
We'd need a better source for the ateji claim. None of the kokugos mark 賽 and 采 as ateji.
abbr was on the wrong sense.
I think the さえ reading can be dropped. Only koj and nikk have it. Presumably archaic.

@@ -6 +6,4 @@
-<ke_inf>&ateji;</ke_inf>
+</k_ele>
+<k_ele>
+<keb>采</keb>
+<ke_inf>&rK;</ke_inf>
@@ -12,5 +14,0 @@
-<k_ele>
-<keb>采</keb>
-<ke_inf>&ateji;</ke_inf>
-<ke_inf>&rK;</ke_inf>
-</k_ele>
@@ -19,3 +16,0 @@
-</r_ele>
-<r_ele>
-<reb>さえ</reb>
@@ -26 +20,0 @@
-<misc>&abbr;</misc>

This is probably the most important piece of info right here:
賽の目	13866	17.2%
采の目	585	0.7%

Do a google image search, and 采の目 doesn't even return dice.  However much 采 might sometimes mean 賽, it is not by any means a common form (its odd blessing as an alternate form in 賽は投げられた being the seemingly-odd exception).

Other forms:
さいの目	37860	47.0%. WARNING: cubed vegetables
サイの目	8044	10.0%. WARNING: "eyes of a rhino" (google images) 犀＝サイ
賽子の目	53	0.1%
サイコロの目	20120	25.0% The modern form wins out over any kanji entry
骰子の目	45	0.1%

Rolling dice:
サイコロを振る	10799	80.8%
賽を振る	1149	8.6%
サイを振る	348	2.6%
采を振る	71	0.5% #采 rare, and this expression means "to wave a baton"

賽を投げた	627	4.7%
サイコロを投げた	284	2.1%
サイを投げた	90	0.7%
采を投げた	0	0.0%  采 does not make it onto the ngrams at all

骰子 
https://furigana.info/w/骰子
50% さい
50% サイコロ

sankoku さい：賽・骰子
「文」：さいころ
＊さいは投げられた...句 (we have this)

Reverso 采+dice (no matches)
https://context.reverso.net/translation/japanese-english/采#dice

Reverso 賽+dice a few results
https://context.reverso.net/translation/japanese-english/賽#dice

https://ja.wikipedia.org/wiki/サイコロ
サイコロ（骰子、賽子）、または賽（さい） 
采　not used, except for a final note at the end(Note, also identifies these as ateji):
「さいころ」の「さい」は、「塞」または「簺」の音読みであり、双六に似たゲーム、もしくはそれに使うサイコロのことである。それに接尾辞「ころ」が付いて、「さいころ」となった。 「采・賽」は当て字である。

yourei baton examples (just looked to see if 采 ever appears there as "dice") 采振りの二人 = two baton-waving individuals
踊る喜びや確かな誇りがある。剣櫂、采振りの二人の呼吸が渾然一体となる踊りこそが見どころである。
Waving a flag or baton after a battle:
敵の本隊さえたたきつぶせば、なんのこともないからだ。「かかれ」 鳶の巣山の戦いにややおくれて、勝頼さまは采を振られた。ご本陣から最前線まで、さほどのへだたりはない。

In short, maybe 5~10+ baton-wavers among the results, no dice.

The gambling game with ピンゾロ (a roll of all ones)... Same as yahtzee- wikipedia articles on dice games just use サイコロ, in kana, which lends credence to this being [form]
https://ja.wikipedia.org/wiki/チンチロリン

Main points:
1. sankoku doesn't give 采
2. wikipedia doesn't list 采 for dice in its intro. (Does list さい・賽)
3. image results return no dice for formulations with 采
4. 采を振る, rare already, doesn't primarily mean "roll dice", it means "wave a baton" 采配を振る (at least most commonly)
5. rather than just count ngrams of 賽 and 采, I have considered them in the context of doing dice-like things (を振る、を投げる,~の目) which confirm that sankoku has a reason to omit 采 entirely.

I think this entry needs to be split.  I've deleted the baton meaning with the idea that someone might re-create it as a separate entry.  I've made this entry look like I think it *should* look for dice.  This includes a single-mora reading of 骰子, given by sankoku and seemingly confirmed by furigana.info. Marked 賽 as [form] following sankoku, but that seems a little strong.  But certainly サイコロ is the common, everyday form.  

Here is a perfect match(and the only image match for 采を振って) on twitter (a mascot at a traditional sporting event waving a baton):
https://twitter.com/kazumin_sh41/status/1581141712159637504
This was meaning [2], for which 采 may be the only applicable kanji (short for 采配). It is certainly the primary kanji for this meaning. I have not looked at this in detail. 

賽を振って　many pictures, half of which actually have dice:
https://twitter.com/search?q="賽を振って"&src=typed_query&f=image

Learners should learn サイコロ. If they want to learn the single-mora form, they should learn 賽.  采 appears to be a point of odd trivia (it does appear in daijs next to 賽).

@@ -5 +5,2 @@
-<keb>采</keb>
+<keb>賽</keb>
+<ke_inf>&ateji;</ke_inf>
@@ -8 +9,7 @@
-<keb>賽</keb>
+<keb>骰子</keb>
+<ke_inf>&rK;</ke_inf>
+</k_ele>
+<k_ele>
+<keb>采</keb>
+<ke_inf>&ateji;</ke_inf>
+<ke_inf>&rK;</ke_inf>
@@ -18 +25 @@
-<xref type="see" seq="1574680">骰子</xref>
+<xref type="see" seq="1574680">サイコロ</xref>
@@ -19,0 +27 @@
+<misc>&form;</misc>
@@ -22,6 +29,0 @@
-</sense>
-<sense>
-<stagr>さい</stagr>
-<pos>&n;</pos>
-<xref type="see" seq="1295140">采配・1</xref>
-<gloss>baton (of command)</gloss>

Entry created