IIT

Any institution whether it is a government or a private institute cannot become prestigious and top institution within a year of its establishment. Also, new IITs have their own identity and cannot be measured with the old IITs. they are not taking as much time as the time that had been taken by the old IITs to evolve into a heritage place for engineering students. The craze for the old IITs is the same as it was before but it's increasing for the new IITs because the curriculum, syllabus and management is just as same as that in the previous IIT but accompanied with a newly developed modern infrastructure. Also, with the increasing number of IITs, the fundings, as well as the facilities provided by the government, has increased so one cannot say that it has diluted the remarkable achievements of the previous IITs.
Thank you.

The answer to this question includes political as well as logical touch although both the institutions are centrally funded. There is a large gap between the quality of students getting seats in an IIT as compared to the one getting a seat in NIT. Also, the government has to maintain a good image for the technical education system of India so the preference is given to the IITs. Although the amount of money does not very much which is around 3500 crores a year. The money issued for officials working in the institution, the fests held and the job perspective is more for IITs again for creating a good image worldwide. Also, the government is not that much capable of getting the NITs and IITs to the same level. The reason is that one needs to understand keeping in mind the political statistics. NITs are doing too good when it comes to education for a better future scope.

Devoting one full year for competitive exam such as JEE preparation has its own merits and demerits. Dropping a year means that you will get sufficient amount of time to prepare for JEE and other engineering competitive exams. It will allow you take mock up test and help to score good ranking in competitive exams. However, one year is a long time so the possibility of you may change your goals and objectives. You may become lethargic because of availability of time and lesser pressure. You may waste time. If you are confident of keeping the momentum, dedication, determination, readiness to work on a plan, will to achieve your objective and support of family; you may consider dropping a year for competitive exams. Doing engineering from IIT can help your career success in long term. It is worth try option.

There are some good institutes apart from IITs, NITs and IIITs. You can check this list:
https://www.shiksha.com/b-tech/ranking/top-engineering-colleges-in-india/44-2-0-0-0. This list also includes IITs, NITs and IIITs. So, I will mention the colleges other than IITs, NITs and IIITs here:
BITS-Pilani
Indian Institute of Space Science and Technology
Anna University
IIST
DTU-Delhi
NSIT-Delhi
Jadavpur University
BIT Mesra, Ranchi
PSG College of Technology
Institute of Chemical Technology, Mumbai
PEC University of Technology
IIEST, Shibpur
TIET, Patiala
HBTU, Kanpur
College of Engineering, Pune
I am doubtful about these but some rank them higher so I will let you know (I personally believe that you should avoid these colleges)
VIT-Vellore
SRM
Apart from them some good colleges which you can go through JEE or KVPY are:
IISC
IISER
Also, check Indian Institute of Statistical Institute if you are passionate about maths.
All the best.