While it's not rocket science it does take some skill and tools and is tedious. Hence the labor charge. I would leave stairs to the experts and try to do something else. If you are installing other flooring, that would be much easier for you to do than the stairs.

I'll admit that charge does seems rather high, so you probably need to speak with a couple more flooring guys. Perhaps the one you spoke with is extra meticulous and intends a spend a great deal of time getting the stairs absolutely perfect. Perhaps the last stair job he did turned into a major adventure and he wants to make sure he's covered for whatever he finds at your job. Perhaps your stairs are unusual construction.

The stairs are the detail which will make or break your hardwood install job. Although there are some stair sanding tools available, there is still a lot of hand sanding and scraping and meticulous work to getting the job right. I saw a stair job once where a couple stairs at the top had corners that were probably 95 degrees (larger angle than a regular corner) and the result looked horrible. Stairs need to be right because you can't hide them behind a door or disguise them with a fancy paint job

I got a similar quote from a reputable company for putting hardwood on existing stairs, in that it was the major portion of the cost of the whole job, and I only had three stairs at the entry to deal with. They are a lot of work, while the rest of the floor goes down quickly. I remember being quite shocked at what it would cost to do the hardwood on the stairs.