It's unlcear if the virus from ENT plays any kind of a role, but the following is one reason why viewers witness [prime timeline] Klingons looking different than other depictions from various [prime timeline] Treks.
"The empire is very big. They don’t all grow up on Kronos. They don’t
all live on the same planets and certainly those different planets
would have different environments. So how would the cultures have
evolved differently?…We tried to come up with cultural axioms for each
house so each looks different and they bear a cultural patina like our
cultures do here on Earth."
"What can you say to reassure us that we’re not losing the Klingons we
know and love?” a furtive audience member asked during the Q&A
portion. Mitchell assured the crowd that the recent publicity still
image released was of one Klingon, from one house. “We will see all 24
houses and the leaders among them,” he revealed. The houses will be
explored, and the physical and ideological differences between them.
L’Rell is part of two houses, Chieffo explained, and the conflicts
arising therein, as well as how she is viewed by the Federation versus
her own people, will be explored in
So although it is made clear that there are various houses of Klingons, and that the more familair versions are then limited to Kronos Klingons in other Treks, it doesn't explain why that limitation exists.
Some speculation on my part is that because Discovery is really a covert cold war story, it could turn out that many of these houses cease to exist (or exist in known parts of space at this time or known times in known Treks, in this timeline) by the time of Discovery's end.