Rec 2020 HLG colors look incorrect?

I am grading in Rec 2020 HLG because HDR looks great on the phone and TVs, but the colors from Rec 709 footage (I think, it's GoPro Flat so probably R709) look over-saturated if not converted, but then red looks ORANGE if the color space is converted but proper saturation is achieved. I looked in the CIE chromaticity chart and it seems what the CST 709 to 2020 is doing is clamping the footage back to Rec709 within the Rec2020 space.

Is there any way to make red look red again when working in Rec2020? I know it's normal for it to look orange on a limited gamut display like my computer monitor but the problem is that red also looks orange on my phone when displaying the footage properly in HDR. Should I just go without conversion and desaturate the footage so that it isn't oversaturated in HDR?

I know the best practice would be to shoot in a wide gamut for a project but that's not always possible with some cameras I use such as the GoPro Hero 9, which is why I believe CST exists, so people with more limited or broad color spaces can do rough conversions to their desired space and to get the colors to look right in that space.

Here is the after the color space transform, as you see the red spike (which is my friend's shirt) is shifting towards yellow a bit.

Meanwhile without conversion, the reds look correct on the vectorscope, but the color is far oversaturated as you can see in the CIE Chromaticity.

For reference, here is the image after the CST.

Here is the image before.