There's allot of variables with your question.
- If it was just a cheep piano I would just leave it, it's not worth the expense. Since it's a Steinway it has value, monetary and perhaps sentimental.
- It looks pretty bad so I don't think I could go with the "leave it alone, it's got character" line of thinking unless you don't really care. Since you posted this thread, I figure it's been bugging you.
- A wide shot with a bigger picture of the affected area would help. From you video it sort of looks like it might be a section that could isolated and taped off and sanded down (at the corners) and new coatings could be applied....... possibly. A wide shot would help determine this.
I've been in custom home construction for 25 years and we install a ton of hardwood into my houses. On a rare occasion a brand new hardwood floor gets some damage and we need to fix it for our customers. It can be tricky to fix a small section of brand new site finished hardwood but it can be done. Just like repainting a quarter panel on a car. A good painter can match the color and finish pretty close but never perfect. An amazing painter can work miracles.
Show us a wide angle and I'll give you my 2 cents..... for what it's worth.
QSS