A registrar takes your money & coordinates with TLD owner to add your domain to the master phone book. In about x time, minutes to hours to few days, all servers over the world has your domain's dns information. Most of the time registrar's job is done, & now hosting provider as well as dns servers come in picture.
Now, when a user types your domain, his internet provider or dns address server looks up the relevant ip & or ask around. Closest/fastest/reliable records are then used. Your registrar usually do not control or operate these servers, but Google 8.8.4.4, Cloud flare 1.1.1.1, your ISP & such operate these.
The geographic closeness to hosting servers is preferred, but I think geographic closeness to register doesn't matter much.
Your understanding is incorrect.
There are no issues with registering a domain via Namecheap when your users are in India (or anywhere else).
Once you've registered a domain, you can use any nameservers that you like -- regardless of their geographical location. You could use the free DNS service Namecheap includes with your domain registration, nameservers from the provider of your choice in India, or something like AWS Route 53, Cloudflare, or Google DNS, which all use anycast and have DNS servers around the globe (I don't know which of them, if any, have servers specifically in India, however).