why should the Yakuza care? There were two hotel owner killed in the neck of the woods, no witnesses, no security system no clues who did it. Maybe she ransacks the office and steals some money while she is at it and it looks like a robbery gone wrong. Who in his right mind would suspect the Yakuza for a robbery in bumfuck nowhere in the middle of the USA?
Apart from the other answere you got.
Here's a couple of things aswell.
1. Etio is a greedy twat but not a tight twat how do I know well he has cameras to spy on the guests.
And as a greedy twat of course he's gonna have a security system incase he's robbed.
2. Etio told them his employee's IE more than one would attempt to hold up Rin's departure.
So there is allways a chance of a witness.
For instance the cousin is safe with Rin and Ryker you think she won't tell the cops it was Yakusa.
3. And this is a guess but how many rural robbery murder's in a country of easy Gun access.
Do you suppose are cut to pieces with a Katana?.
And finaly Apart from the fact that Etio is able to point them in the right area.
They really still wouldn't want to risk Rin hearing about a double murder at the place she just left.
And when She did they would assume that Rin would once again be gone with the wind.
I should just like to add these assumptions of mine are based on the belief that.
The Yakusa maybe Ruthless Killers but their Not Stupid Ones.
Just sayin.