Surgery Clinic Quoted Bill Thousands Less Than Actual Bill

Looking for advice and seeing if I have a case. I am in the U.S.
I had a surgery completed last year. The procedure was unsuccessful. I provided the surgery center my insurance information. After reviewing my information and contacting my insurance, the Surgery Center quoted me a price under $1,000 which I paid prior to the surgery. I have now received a final bill from my insurance. The bill is for $9,000+. The insurance is not covering the most vital procedure in the surgery.

I have reached out the physician's clinic to confirm the correct billing procedure code was used by the surgery center. I am waiting to hear back.

My questions:
1a) If the surgery clinic quoted me the price after having my insurance information, are they responsible for the extreme discrepency between the quote and the actual bill?
1b) Is there a legal limit on the price difference between a quote and a bill for which the patient is responsible?
2a) If the surgery clinic is not responsible for the price difference then what else can I do to mitigate the bill?
2b) If the surgery clinic is not responsible for the price difference, what is the purpose of the quote?