Other than the two most powerful legislators' being arrested....

Wow, Ben, that's really a good q. There's an implication of "...than the matter of the two..." where "legislators' " modifies "matter." But it's a stretch

We use the possessive with gerunds (because they are nouns):

My being arrested
Your paying for my meal

When using any possessive apostrophe it goes before the 's' for singulars:
John's being arrested
The man's hat was black.
The legislator's being arrested was a shock for his family.

It goes after the 's' for plurals:
The mens' hats were different colours.
Two actresses' roles
Guys' night out
The legislators' being arrested was a shock for their families.
It can sometimes also go after a final 's' even for a singular noun, although this is disputed.

Jesus' name
Thomas' house

OK, I know about putting the possessive apostrophe, and all your other examples make sense, but I don't understand why
...the legislators' being arrested...
is possessive.

We use the possessive with gerunds, and being is a gerund.

For one legislator we could write about
His car (The legislator's car)
His wife (The legislator's wife)
His age (The legislator's age)
His being arrested (The legislator's being arrested)
His being stupid
His being drunk
etc

For multiple legislators we could write about
Their cars (The legislators' cars)
Their wives (The legislators' wives)
Their ages (The legislators' ages)
Their being arrested (The legislators' being arrested)
Their being stupid
Their being drunk
etc

What don't you understand?

I do not think it is.

Even the NYT can be grammatically incorrect at times.

Frank, it is grammatical.

I will concede that either way (with or without) is grammatical...and acceptable, Tes.

Tes,

OK, I think I get it. I just looked up "gerund" in Wikipedia, and their example

"The teacher's shouting startled the student."

make clear why the gerund takes the possessive. For some reason, "being" threw me.

Thanks.