The vaccine underwent Phase I and II safety trials, and was thought to be safe. Russia than began directing the limited quantities being manufactured towards at-risk health care workers, while they ran efficacy trials.
There's nothing magical, or ordained by God that says we must run an efficacy trial with 30,000 individuals. By early January, Russia had only administered a couple hundred thousand doses.
In retrospect, it's highly likely this would have been the correct approach in all vaccines so far in use. In the middle of a pandemic, there's a very strong argument that should be vaccinating as quickly as possible after Phase II trials are complete, before waiting for a long-running efficacy trial. As you said, efficacy trials are not testing for safety. There's already a huge risk to individuals and to society from the virus itself.
If vaccines were widely deployed after phase II with no proof they actually work, there would have be some that did not work that got widely used. The people that got them would they have immunity, and slack off on distancing and masking.
Once you did have a vaccine that you knew worked, there would probably be problems convincing those who got an ineffective vaccine that they need to come in again.
FDA is not a universal regulator. Different countries can have different procedures.
phase 1 : 1-100 phase II 100-1000 phase III 1000-3000 https://d18x2uyjeekruj.cloudfront.net/wp-content/uploads/201...
They held out for something else because the Russian vaccine had undergone no phase III trials yet. It had only undergone phase I and II trials, which just test whether it is safe to give to people, not whether it actually does anything helpful.