The term 'popular' always comes with a caveat, doesn't it? Like, do we mean mainstream hit or a cult favorite the fandom won't shut up about? For a romantic core that genuinely drives the whole story, I'd point people toward 'Clannad'. It's a classic, and for a reason—the After Story arc especially builds this profound, heart-wrenching family love story that goes way beyond high school fluff. It's emotionally exhausting in the best way.
Another one that consistently gets recommended is 'Steins;Gate'. Okay, hear me out—it's a time-travel thriller first, but the relationship between Okabe and Kurisu is absolutely central. Their bickering-turned-deep-bond fuels half the plot's emotional stakes. The romance feels earned because it's woven into the survival narrative, not just tacked on. I see more people getting into 'Fate/stay night' for the fights, but the romantic routes, particularly the 'Heaven's Feel' path, are intense and defining for the characters, even if they're wrapped in a Holy Grail War.
That said, the 'popular' list always includes things like 'Katawa Shoujo' (free and surprisingly nuanced despite its premise) and 'Doki Doki Literature Club!' which... well, it uses romantic tropes as a vehicle for something much darker, so maybe that counts? I'm less sold on 'Nekopara' being a 'strong romantic storyline' unless you're really into catgirl antics over plot.