Higher-order interactions, which generalize pairwise interactions to many-body interactions, are gaining increasing attention in various fields of science, but explicit real-world examples are still not so common. We show that, in Kuramoto-type coupled phase oscillator models, higher-order interactions naturally emerge from time-delayed pairwise interactions commonly observed in real-world coupled systems. Furthermore, we demonstrate, both numerically and analytically, that the higher-order Kuramoto model captures the dynamics of the original time-delayed Kuramoto model more accurately than the conventional pairwise Kuramoto model based on the Ott-Antonsen ansatz (OA). As an application, we control the time-delayed Kuramoto model and show that the OA equation for the higher-order Kuramoto model provides more accurate predictions.