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I did 尻取り games when I was a child before the war—more than 70 years ago—as every child did.
尻取り was one of the most popular and inexpensive games played among pre-war children, because they didn't have video games or smartphones to kill time as today's children do.
I and other children never paid attention to whether the ending letter of the word was を or お, え or ゑ, へ or え, because we were unable to tell the difference of the usage of these words which are pronounced same but written differently. We linked the words only based on the ending sound (the last syllable) of the preceding word and the beginning sound (the first syllable) of the following word, not on the characters, when playing a 尻取り game.
There was a problem, which occurred when the ending of the precedent word was ん, as there is no Japanese word that starts with ん. In that case, we followed the previous word with うん such as 運転手、運動会, and sometimes, うんこ and うんち, as we didn't know then big words like 運命、運気、運否天賦、薀蓄.
By the way, I'm curious to know whether Anglo-Americans, who don't have syllabic characters as we have in あいうえお, have a similar word game to 尻取り or not, and what they call it if they have one.
You already knew that しりとり=尻取り, hip takings, am I correct? ( But let me answer since I did not at all. )
If you google by the word 文字鎖，a site comes up..
There 武者小路実隆｛むしゃのこうじさねたか｝（1661～1738), read 源氏物語，according to the rule of 文字鎖，whose pronunciation of the end of each verse corresponds with the pronunciation of the head of the next verse. Like this.
Apparently, you can easily guess the origin of the word, しりとり is 尻取り， which I guess 尻｛しり｝, hip, in turn, would mean the end of the each verse, and might be connecting it.
And I guess the word play has been passed on until the end of the war,
According to here, something like しりとり is read or said,
So I guess, "people" did not use the old readings? when they play this word play???
Thank you again for your interesting question.