I think it depends what you do during that time.
If some countries had gone into lockdown a week or two earlier, but still spent resources ensuring hospital supplies, and working on producing tests, and contact tracing, and developing a vaccine, then the initial peak would have been lower, and you'd be in a better position earlier to move into the 'mopping up' stage.
Yes, you'd still have the risk of isolated outbreaks that would need to be 'mopped up', and you have decisions to make about incoming and outgoing international traffic and mopping up small outbreaks that crop up due to incoming traffic.
Surely it would cost less money in the long run, and lives. So I do think it would have been worthwhile. That's part of the reason why it's been so devastating: because it was new and we were unprepared. We have so many more things in place now that we didn't have back in February/March (in the UK and other countries that have been on a similar Covid-19 timetable).
When it first 'started' in the UK (meaning when we noticed it starting) we were only able to test a small number of hospital patients. Now we're in a position where we can test 100,000+ a day, and anyone with possible symptoms can request a test. So if we could have reached this position without having such a high first peak, I don't see a downside to that.