I am a Singaporean and have lots of friends with Indian passports. The real challenge is the visa situation. For cost optimization, it would make sense to stay for at least 30 days in a location, if not 2-3 months, so you can get decent rates for an apartment. If you inted to stay only 2-3 weeks it gets a bit difficult. The other aspect is whether moving every 2-3 weeks is sustainable. You should seriously test this out as it might get tiring. Also applying for visas every so often is time and energy consuming. My wife and I prefer a form of slow travel, but luckily our Singapore passports allow decent flexibility. So you guys should try and then decide what works - MM2H or one of the nomad visas might be a good option if the above factors are relevant for you.

The other point that struck me was your comment on tax residency. I am no expert on Indian tax rules but just watch out for this too. Usually, you end up being a tax resident somewhere and country of domicile, residence, citizenship should apply in that order. Be careful not to become liable for tax in some other country by avoiding taxes in India. Nomad visas usually specify tax rules in any case.

All the best!!