Let me preface this by saying, yes, Rugby Union has established a fairly decent international presence.
I would stop short of calling it a global game though. They are not as big as they'd like to think they are on the world stage and Australia being absolutely crushed is not good for their world game. If NZ and the rest of the Pacific fall to RL too, they all of a sudden have lost a few key pieces off the chess board.
1. The All Blacks: No Union brand transcends the sport like the All Blacks, if they follow the Wallabies' lead and become mediocre, helped along by an aggressive NRL push into NZ, they lose something unique the sport had. Yes, it is popular in Ireland, Wales, pockets of France, southern England and South Africa. But none of those countries has Union running in the veins like it does / did in NZ and the Pacific Islands and that is a huge blow.
2. Playing talent: Not unlike league, NZ and the Pacific is relied upon for a disproportionately large percentage of their global playing talent (for just about all nations). As the famous line from the commentator goes from the 2023 RUWC... ''and another Kiwi scores for Ireland''. If this talent increasingly starts choosing league, they are left to rely more heavily on South Africans, English and British talent and it just isn't as abundant as the game is not as deeply and culturally ingrained in those countries as it is in the Pacific.
3. World Cup Quality: If you are claiming to be a global sport, only have 10 professional nations playing and two of the historically most competitive sides (Aus & NZ) are suddenly mediocre and unable to compete for major titles, it suddenly leaves you with a shallow pool of England, France, Ireland and South Africa that are any realistic chance of winning trophies... Is that much different to the ''tiny'' sport of Rugby League where You have England, Australia and New Zealand as the big dogs?
Anyway... cheers to the decline of Union.