Is the use of JEE score only to get a seat in an NIT? I got 77.678 percentile in JEE first attempt.

No, apart from getting seats in NITs, IIITs, GFTIs. It is used to select the candidates for JEE Advanced to get seats in IITs and some colleges and universities accept JEE scores to give admission. So, JEE score has many uses if you get 100 above score.

Please visit this site first- https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor&ved=2ahUKEwjd0NzModzhAhUZfSsKHfLtA6UQFjAAegQIBBAB&usg=AOvVaw1N9BzFtyUCP4JU-2UQHQ0H.

Not many other government private colleges such as Thapar University, Jamia, AMU, etc.

No, JEE scores are accepted by many colleges and universities across the nation. You can get government colleges of Rajasthan, Punjab, Delhi and more. You can get CFTIs and many private colleges like Thapar, Jaypee Noida, etc.