I didn't know they made engineered floors finished on site.

I have solid hardwood, 3/4 select white oak, stained and finished on site. when we made the decision we ewre between site finished and prefinished. at first my wife was piushing for the prefinished b/c she thought it would be significantly faster, we weren't living there at the time. it really isn't for a good flooring company. All the wood was down in 1 day, and another 3 of finishing.
Price for the oak wasn't significantly different either way, maybe 5%.

Either or, I like the seamless look and feel, it was what I grew up with as "normal" so that may be why. I know a few people with prefinished, sometimes it looks/feels fine, sometimes not so much.

If I fell in love with a prefinished product that had a special finish applied, ie some of the handscraped stuff, the exotic woods that have some real interesting coloring etc, I would go with that. Otherwise I prefered site finished. Again though, for me the difference was 5%. at 50% difference I might have a different opinion

Yes - if there were a 50% difference, that would make the decision for me!

Since there wasn't a large price difference, we went with pre-finished because we were living in the house and did not want to live through the dust and smell.

That was five years ago and we've been pleased with the result. The boards do move a bit from summer to winter, but so do the ones in our downstairs that were sanded and finished before we moved in.

Both upstairs and downstairs, though, we have hardwood, not engineered. Well, downstairs we inherited yellow pine and even our 22 pound Beagles scratch it! Upstairs we have the pre-finished hickory and that's held up very well to the three critters + their humans.

Of course, YMMV.