While reading your post, I thought that you wanted to say that today S1590 is raising an issue on DELETE statements using INNER JOIN. So I tested it and so far so good, there is no issue raised by S1590 on such statements (at least your sample code).
Do you have on your side using the latest version of SonarTSQL (1.3.1 as of now) an issue raised on this statement?
DELETE p
FROM Products as p
INNER JOIN DeletedProducts as dp
ON p.ProductId = dp.ProductId