Which should I choose, mechanical engineering at new IITs or engineering design at IIT Madras?

Engineering Department usually has more companies visiting for placements than IIT Roorkee and the condition is worse in new IITs. The average package is also at par with the packages of most branches.

In fact, any branch at an old and big IIT like Madras is better than CS at new IITs. the packages of CS at new IITs are on the same level as packages offered in normal branches of old IITs. Moreover, the campus life and exposure offered in big old IITs can never be compared to the new IITs which are still struggling to reach IIT standards in terms of their infrastructure and placements. 

Secondly, Engineering Design is not completely mechanical, but rather a combination of electrical and mechanical. This department has the highest number of labs and assignments. and thus a lot of assignments.

Overall, the coursework pertaining to the department is full of rigor and requires much more effort than other branches. If you are ready for that kind of commitment, go for it but if you eventually want to do coding, chose an easier branch that will give you enough time to learn to code.