The phase-field method has been widely used to study the rapid solidification processes such as additive manufacturing. To lower the computational cost, many simplifications and modifications were made to the original set of phase-field equations for alloy solidification. However, how those simplifications affect the modeling outcome is not well studied. In this work, we developed a phase-field model incorporated with both the coupled thermal-solutal diffusion and solute trapping for the rapid solidification of dilute binary alloys. We focused on the quantification of the influences of the simplifications of interface kinetics, solute trapping, and thermal diffusion on the modeling outcomes. By turning on and off those three effects individually, we explore how the temperature, solute concentration and interfacial velocity are affected. Our investigation shows that the interface kinetics and thermal diffusion cannot be ignored for quantitative modeling of a rapid solidification process for alloys. In the end, we used the rapid solidification of a melt-spun ribbon as an example to show that the phase-field model with all the three aspects considered is capable of capturing the complex interplay among temperature, solute concentration, and interfacial velocity and predicting the resultant solidification structures.