IIT

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.