One of the ways in which the Remake really shines is the agency and independence it gives to both women. Cloud is important to Tifa, but he's not the only thing in her life. She's dedicated to Avalanche; immediately after losing Cloud in the Airbuster battle, she throws herself into the Don Corneo mission. It's reminiscent of her line in the OG when she says something like, "Right now I feel I have to push myself to the limit." There's even a scene in which she is shown prioritising her Avalanche commitments over Cloud (and he doesn't really like it).
Aerith knows more this time around. She's aware that she, too, has a mission, and that it's much more important than any romantic interests she might pick up along the way. She may wish it wasn't that way, but it is.
As the player our focus is on Cloud, but we shouldn't forget that both these women have their own stories, of which Cloud is just a part. They don't live and breathe and move only for him. He rescues both of them from dangerous situations, and these rescues form a large part of the plot and the fun. However, they are also both shown rescuing him*. Sometimes they work together to rescue him. Tifa and Aerith have a rapport of their own, independent of Cloud. Aerith is good at breaking down everyone's defences; it's what she does. So the fact that she gets through to Cloud isn't per se proof that he is special to her (although of course we know he is, or he comes to be). The game designers repeatedly and deliberately show us Tifa and Aerith collaborating together, supporting each other, and bonding.
After all the work they put into developing both these women into fully fledged protagonists in their own right, and friends to boot, it would be pretty crass to pit them against each other over a man. Each woman in her own way is out to save the world. They don't have time for a love triangle!
* Edited to add: each woman is both vital in the other's rescue. Cloud couldn't have rescued Tifa without Aerith's help. He couldn't have rescued Aerith without Tifa's help.