No, this is not true in general. There are several different ways to count the number of covid deaths, and "had covid, later died" was simply one that was often used at the start of the pandemic of efficiency reasons.

Good investigators have always been aware of the limitations inherent in the assumption, but initially other measures based on the official death certificates were simply not available (and in any case have their own complications when interpreting), so people used what was available and cautioned about the limitations the first four times they said anything.