Side effects such as fever and chills were more common in people who got the new vaccine.
This caveat, glossed over in the article, is potentially a huge drawback.
Personally, the covid and flu shots by themselves make me feel icky but moderately functional for 24-36 hrs, but the time I had both at once I was practically bedridden for three days.
Obviously not everyone has this problem, but from what I've heard it's common enough that I worry if the two-in-one shot becomes the new standard it will turn a lot of folks off from vaccination altogether. There's enough vaccine hesitancy and skepticism out there already that a miserable reaction (or simply the need to clear your schedule for three whole days) could be enough to cause folks to indefinitely postpone their yearly booster or forgo vaccination altogether.