In this hypothetical scenario, who gets to decide how the body double should act or what decisions he takes? And Why wouldn't they just take the power themselves?
The latter because as soon as it’s know Putin isn’t there a power vacuum is evident and a broader struggle for power ensues. Then whoever wins has to establish legitimacy.
Whereas a few people in agreement about how to direct a widely-recognized figurehead can hang on to power indefinitely and without contention (the latter being potentially costly along many dimensions).