This is a problem on a lot of HDR-capable consumer monitors, even expensive factory calibrated ones from Dell. They often lack any form of sRGB clamp resulting in exactly this problem when viewing an sRGB signal. AMD has a setting in their GPU driver control panel that can do this on the GPU without needing the user to provide an ICC profile. The Nvidia driver has a hidden API to do this, requiring the use of third party software to take advantage of.[1]
TVs are a little better in this regard. My Samsung TV has an "Auto" option for color gamut that correctly identifies DCI-P3 and sRGB signals and clamps them accordingly. But the default setting was "native" which made sRGB content look really bad.