Okay, I found a way to approach it. PCS has a list of the riders with most stage wins in Grand Tours (annoyingly they have a flaw which makes you unable to view more than the top 50) so I checked the top runners.
With a total of 64 stage wins, Eddy Merckx has won the most stages in the grand tours. He won 24 stage in the Giro d'Italia, 34 stages in the Giro d'Italia and 6 stages in la Vuelta a Espana.
www.procyclingstats.com
It turns out that Rik Van Steenbergen and André Leducq won in all their 9 Grand Tours.
Rudi Altig won in his first 8 Grand Tours.
Alfredo Binda, Gino Bartali and Giuseppe Saronni won in their first 7, so Roglic is equal to them.
So, possibly a joint 6th place so far, but I didn't check beyond the top 50. Possibly, all who have at least 7 stages could potentially be equal to Roglic in this statistic and even though one can sort by country and in that way find everyone who fulfills that criterion, there is a limit to my interest in this matter, so I decided not to do that
Other active riders who are in the running for this are Pogacar, Van Aert and Ackermann with 2 Grand Tours as well as Hirschi, Ganna, Jakobsen and Damiano Cima (!!) who have all ridden just one Grand Tours and won at least one stage.