I got my camera back yesterday (Friday).
The service write up claims they replaced the shutter mechanism, they listed "related parts" 6x underneath the line stating they replaced the shutter, they cleaned the sensor, and performed a general check and clean.
Some of the first photos I've taken were of a perfectly blue and very clear sky (no clouds, nothing). Before I put the lens back on, I tried to make extra sure that the rear element was free of dust and dirt and marks, and I did the same for the front element before taking shots of the sky.
Zooming both 1:1 and 2:1 on the photo and checking everywhere, I see no dust spots or other dark spots of any kind.
I also pulled up the shutter actuation numbers. It looks like they performed about 2000 shutter actuations.
Hopefully the large majority of those were on the new shutter. I only had about 3400 actuations before I sent it in, and I had stubborn spots that matched the description of the oil splatter. If they performed nearly 2000 on the new shutter, I'd be seeing them now or very soon.
It's still possible that they had spots after those 2000 actuations but after a clean it's good - I'll keep firing away and every now and then checking against a nice background like a blue sky to check over time, but so far so good.
I'll admit I was hoping to get a D610 back in exchange for my troubled D600, but I'll accept this camera performed as accepted.