Firstly these "fact check" websites, including Reuters have become a joke.
They clearly have an agenda and many also have conflicts of interest.
They pretend that just because they call themselves "fact checkers" it automatically makes everything they say true. They have nameless faceless self-proclaimed "experts" that often make a straw man and "fact check" the straw man as false and call it a day.
And a naive person just googles a few keywords, sees the - usually first link - "Fact-check: that thing you were told is totally wrong" article and thinks "Hah! Fact checkers say it's false! Debunked misinformation again!"
Big pharma has billions of dollars of interests to protect. The average so called "anti vax" person has nothing to gain or lose from whether other people do or don't take a vaccine. So a rational person should not so easily accept the commercial propaganda pushed by these companies.
Secondly health care professionals submit the report when they have reason to believe that it is a "vaccine adverse event". Otherwise the "event" is not relevant, the person will not even consider or bother to make the report to begin with.
"even if it is unclear whether the vaccine was the cause" part is a technicality because yes at the time they are submitting it "is not clear" if vaccine is indeed the cause and it is to be investigated.
They will then contact the person that submitted it and if it's a valid report they will assign a permanent number to the report.
There are many obstacles to submitting the report. And since it is voluntary it is under counting.
In Australia 9 deaths are officially accepted as vaccine-related. Again unlike with covid they do anything they can to keep this number low. It is a minimum.
I don't claim to know what the actual number is but it absolutely positively is not ZERO.