It has something to do with the logistics as well. The vaccine from Pfizer/BioNTech needs to be stored at minus 70 degrees and not too many facilities in India have the capability to do so. The vaccine from AstraZeneca/Oxford on the other hand can be stored using normal refrigeration. The same goes for the versions from Novavax and J&J. Apart from the vaccines approved internationally, India is also using one of the indigenously developed vaccines as well (Covaxin from Bharat Biotech). Currently around 80% to 90% of the shots being administered are Covishied (AstraZeneca/Oxford) and the remainder is of Covaxin. And till Saturday, the cumulative number of individuals who have received at least one dose of the vaccine stood at 5,803,617 (which is around 0.4% of the population).
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India was relatively clear that Pfizer wasn't inclined to do the kinds of tests the Indian govt wanted (on India populations...every population has prevailing genotype issues, common regional infective threats, nation-wide nutrition and sanitation profiles, etc.) If a government is doing their job right, they would be requiring such diligence. That doesn't mean it was the whole story behind the rejection of course.
(Note that rotavirus vaccine works poorly in Indian babies because the mother's milk is particularly potent in that country. The technocratic solution:
Don't breastfeed so that this one particular vaccine can work...and Dr. Paul Offit can get his royalties.)
AstraZeneca and J&J work by inserting custom DNA into the cell nucleus using viruses and letting the natural processes of the cell produce mRNA programmed to make designer proteins. I'd want to see decades of real-world observations before taking such a thing. Also, and very importantly, I'd want an independent analysis (and with adversarial independence) of every bit of genetics in every batch of 'vaccine'. To say I don't trust J&J, who sold AIDS tainted blood products to other nations so they didn't have to dump it, just got found guilty of making carcinogenic baby powder, etc) is a VAST understatement.
Covaxin is plain old inactivated virus. Even given that all such efforts (coronavirus vaccines) over the past 20 years killed the lab animals in startling numbers, I would still consider it the 'safest', and it looks to me as though all countries will have a small number of 'old school' vaccines. I suspect for their 'elite' to use and still get their 'immunity passports' in case they are not elite enough to buy a private jet.
Noravax is a 'subunit' vaccine. This is like inactivate, but it just makes antigen parts in the lab/factory (rather than re-programming one's own cells.) This technology seems to me like it might hold promise for safer vaccines...which is all most of us so-called 'anti-vaxxers' really want.