Reports of critical current ($\rm I_c$) suppression during cryogenic ion irradiation of REBCO tapes have raised concerns for the operational margins of fusion power plant (FPP) magnets. However, the data remain inconclusive regarding beam heating due to the difficulty of measuring local temperatures with contact probes. This leaves a critical knowledge gap concerning the mechanism behind $\rm I_c$ suppression, and whether the so-called \textit{beam on effect} is to be expected under neutron irradiation during FPP operation. In this paper, we show that $\rm I_c$ suppression is independent of atomic displacement rate in the REBCO layer, the latter of which increases twelve-fold as we reduce the beam energy from 2400 to 800~keV. At fixed power, we observe statistically identical suppression with 150~keV protons, which do not have enough energy to reach the REBCO layer, refuting hypotheses about beam on effects being caused by nuclear displacements or direct ion-Cooper pair interactions. These results show that REBCO temperature rise alone can explain $\rm I_c$ suppression, leaving little to no margin for alternative mechanisms. With this insight, we developed a method to measure beam spot temperature that does not depend on the specific installation of our temperature sensor. With this new method, we measured the temperature gradient across the tape during irradiation and found that thermal resistance at the tape/target interface is the controlling variable in $\rm I_c$ suppression. As such, accelerator-based facilities aiming to reproduce the operation of REBCO magnets in a nuclear fusion environment should find strategies to minimize interface thermal resistance. Most importantly, we find that the dose rates expected in a FPP will not change $\rm I_c$ due to ballistic radiation damage or ion-Cooper pair interactions, allowing us to safely ignore these effects when designing FPP magnets.
