My understanding is that if colors appear washed out, it is due to broken conversion between partial and full. Not the inability of a conversion between the spaces. If a display on a computer uses partial color space, and the video driver knows this, it converts any full material to partial and the display displays this perfectly fine. There is a well-defined conversion between full->partial, and partial->full as well, so if a converter knows which format source and destination is, there is a perfect conversion in both directions. Never washed out colors.
Washed out colors happen if given video data is actually partial, but the receiver assumes it is full. The other way round, given video data is full but the receiver assumes it is partial, results in drowned-to-black dark parts and blended-into-white for bright parts.
A converter can be the media player (it converts from the video color space to the computer color space) and the video driver (it converts from the computer color space to the display color space).
If the video driver thinks it has a full color range monitor connected, but the monitor is actually partial, wrong colors will be the result. And vice versa. For example, I have a native computer monitor that only processes full color space. Everything looks correct according to the color space test charts. In Nvidia control panel, I find a setting to activate "partial" output. It's the opened drop-down list at the bottom, the choices say "Full" and "Partial":
Now, if I choose Begrenzt (Partial), the colors of my desktop (including background picture) actually turns washed out. The explanation is it's because the monitor always assumes the signal "full" color range, but it is actually given "partial" output. The driver converts to partial, but the monitor uses this as full and don't convert back, so the colors become washed out.
Usually, you don't need to care about all this. Usually, systems configure themselves correctly. Usually, you just plug in a monitor and it is driven correctly. If you change something manually, it might become a source of confusion, because not every part in the video production+display pipeline is able to automatically detect the color space property of what it is receiving and what it is outputting. It assumes defaults instead. This includes the Full/Partial setting of OBS: Usually it's best to just leave it at its default (Partial).