Google use 200+ signals to rank website and domain name or extension will impact only 0.1%. Sometimes it hurts as well, if it look's spammy.
So Do you think you should gives more time on domain name rather than on other things?
Google main abstract is rank good quality website on top, and keyword in domain name is not a good factor to consider it. Will you consider that if you own any search engine?. Still Google may favor 0.1% if keyword is there on domain, but it is not future of good web, Google already giving less value on that with time to time.
As stephen said, if you go with .in extension, then that site is more visible to Google.co.in because Google think this site is specific to Indian visitors only. You can also use .com and use India as targeting option in Google search console tool. Both will do same things. Regarding .hospital extension, then Google will treat same as other TLD.