I see no evidence to support that claim. They could have imposed much stricter measures to limit public gatherings and do it faster, even more public messaging

I see no evidence to support that claim. They could have imposed much stricter measures to limit public gatherings and do it faster, even more public messaging, clearer guidelines, working to make sure that everyone was inline with the guidelines. Yes it would have helped with massive amount of testing, which they said wasn't possible, and tracing and more effective quarantine.