I think it's popular because it's got a lot of furniture you can use for it. Heck, when I started the game I made my main room into a "granny room", it's not my style at all but all the furniture I had at the time that matched was like, wooden furniture with a lacy rug and a rocking chair... floral wallpaper... I think that's the same type of stuff people use for cottagecore
Part of it's popularity is the furniture selection lends itself to that aesthetic, and part of it is that cottagecore is popular with the artsy, LGBT+ youth who make up a big part of ACNH's playerbase. It's overused in the sense that it's the prevailing theme I see with islands made by people who got the game at release and didn't want to wait for major updates to finish their islands.
As mentioned above, "granny" is not a bad summary of what goes into Cottagecore aesthetic, which aims for a cozy, naturalistic vibe, full of woodsy charm, earth tones, and a sense of clutter. That there are many different furniture that suit it isn't the whole story, though that's a big one; It has a lot to do with the fact that the group of people who like Animal Crossing, a game which loves a nature-filled idyll of little cozy homes and a leisurely pace, and the group of people who thing cottagecore would be a fun island theme, which produces a nature-filly idyll of little cozy homes, form a pretty overlapping Venn diagram.
That one particular path is so popular right now because it has the vibe of old grass decay from earlier games more than the dirt pathing of the game (and, having done natural paths in ACNL, I do just like the look), with an untidiness, earthiness, and simplicity that's all appealing for the aesthetic.