Why vary עלינו?

Question

I took a look at copies of the Baladi Yemenite weekday morning and evening prayer services recently. I was surprised to find instructions to vary the wording of "Alenu" for "בעשי״ת", which I took to mean "during the days between Rosh Hashanah and Yom Kippur", though I'd appreciate a correction if I'm wrong. Specifically, the usual text was

‏… שלא עשנו כגויי הארצות ולא שמנו ככל משפחות האדמה שלא שם חלקנו כחלקם ולא גורלנו ככל המונם שהם משתחוים להבל וריק ומתפללים אל אל לא יושיע ואנחנו משתחוים…‏

… that he didn't make us like the nations of the lands, and didn't set us like all the earth's families, that he didn't set our portion like their portion, nor our lot like all their masses, that they bow to nothing and emptiness and pray to a god that will not save, but we bow…

but the בעשי״ת text was

‏… שלא עשנו כגויי הארצות ולא שמנו ככל משפחות האדמה שלא שם חלקנו כהם ולא גורלנו ככל המונם שהם משתחוים להבל וריק ומתפללים ללא יועיל ואנחנו משתחוים…‏

… that he didn't make us like the nations of the lands, and didn't set us like all the earth's families, that he didn't set our portion like them, nor our lot like all their masses, that they bow to nothing and emptiness and pray to what will not help, but we bow…

(emphasis supplied in each case, and the translations are mine).

Why the variation in the text for בעשי״ת?

Answer 1

The Baladi Yemenite commentary Etz Hayyim by Yiḥyeh Salaḥ writes on Aleinu as found in maariv of Rosh Hashana, and explains the change as follows:

ואומר במקום כחלקם, כהם. והטעם כתב בעל תולעת יעקב, מפני שעדיין לא ניתן להם חלק, אבל בשאר ימות השנה, שחלקם מבורר, אומר כחלקם. וכן אומרים עתה, ללא יועיל, כי, אל, הוא לשון כח וחלק שניתן לשרים שלהם, ובראש השנה לא חלקום עדין, אבל שאר ימות השנה, חלקם נתון להם, והם אל לא יועיל, אין בידם לשנות חלק הנגזר להם, עד כאן לשונו.

And we say כהם instead of כחלקם. And the author of Tola'at Ya'akov wrote that the reason is because their lot is not yet apportioned, but in the rest of the year, when their lot is clear, we say כחלקם. And similarly we say now ללא יועיל, because אל is the language of power and the lot that is apportioned to their leaders, but on Rosh Hashana they have yet to be apportioned, but in the rest of the year, their lot is apportioned to them, and their power does not help them, it is not in their power to change their lot that was decreed for them.

I found this in the tiklal set התכלאל המפואר אור מהרי״ץ. I'm guessing that the Tola'at Ya'akov is the kabbalistic commentary on the siddur by Meir ben Yehezkel ibn Gabbai. Aleinu as found in Rosh Hashana musaf in that tiklal also has the adjusted language (כהם and ללא יועיל). The weekday tiklal from the same set instructs the language change "בעשרת ימי תשובה", so you read the instructions correctly.