The death rate in Italy was in the double digits during the period of their hospitals being overrun last spring. Then they initiated heavy lockdowns, and got their death rate back under control.
Early on, like in March of last year when Italy hit their double digit death rate, CFR was very meaningful. It's only when the breakdown was so bad that they couldn't meaningfully trace anymore that CFR stopped mattering.
Italy stopped having meaningful Covid tracing well before their first detected Covid-related death in February last year. They'd reported zero cases in the weeks leading up to it and there are a whole bunch of reasons to suspect that was a massive, massive underestimate. (I think they also imposed measures like school closures pretty soon after this. It didn't seem to help much.)
