Responses

They never stated that Zigril was the one to kill Kay's family tho.
He mentions the fortresses he stormed and personally had executed, but estates are not fortresses and Kay says that his family fell and were executed for it.
So either they were framed for a crime, actually committed a crime or potentially gained too much power and the previous emperor had to have them culled. Either that or the citizens that Kay's family were ruling over revolted and that's how the family came to ruin.
Either way, they don't ever explicitly state that Zigril was the one to kill Kay's family.
Zig: “lalala yeah I beheaded your family lalalala now marry me and have my kid”
Kale: *blinded by his face card* “o-ok”