To expand on this a bit, if you want leverage over companies like MasterCard, Visa, and American Express, it definitely helps to have the data in your jurisdiction. As you noted, they will shut off mobile internet to support the state. If Visa decides that it doesn't want to give the government data on someone's transactions, it helps to have that data in the country.
"Oh, but the data is encrypted." Sure, but the country could literally seize all of the servers and shut down the payment processor. "Hey, American Express, if you don't give us the data we want, we'll simply take your whole payment network offline."
It's a lot harder to pressure a payment network when you don't have physical control - especially if that payment network can challenge you in court. It's easy for the government to shut off mobile networks that are physically present. It's harder for the government to stop traffic from routing to a payment processor whose servers are outside of the country - especially if they have a half-decent security/threat team ready to avoid blocks.
I don't know the laws of India, but it seems likely that the Executive has a certain amount of discretion that can be challenged. However, it's a lot harder to challenge that discretion if your business is offline for a year while you challenge it. Payment processors need to comply with local laws, but if you're a country where the Executive doesn't mind disrupting services, that gives them a huge amount of leverage to get your compliance beyond local laws. If the data is hosted outside of India, it's a lot easier to to simply comply with the laws rather than the laws plus whatever the Executive thinks they can claim the laws support.
Ultimately, if the data is stored within India, the Indian government can be an existential threat to their business in the country beyond what the law allows. A company will hand over data rather than seeing its servers seized for a year while a trial ensues. During that year, they'd lose all their customers to competitors and they'd be shut out of the market. Likewise, what if the government "accidentally" damages the data when they lose the trial? If you have 100M customers in a country and each is carrying a balance of $100, that's $10B that people owe. If the government has seized all those records and then damages them, your business is in a lot of trouble.
Data residency gives a government willing to bully companies a lot of power. It's simply a lot harder to access information stored abroad and you have a lot less leverage.