Flat is not HDR
This is the mistake worth correcting first. GoPro's flat colour profile produces a washed-out, low-contrast image that looks superficially like ungraded HDR footage, and it is not HDR. It is an SDR curve chosen to preserve grading latitude within the SDR range, usually recorded at 8-bit.
A flat profile and HDR solve different problems. Flat keeps more of what the sensor saw available for a colourist working in BT.709. HDR extends the range the delivered file can carry. Applying HDR signalling to a flat SDR file does not turn one into the other.
Which of your clips are actually HDR
Recent GoPro models added HLG recording in specific modes. Whether a given clip used it depends on the model, the mode and the settings at capture, so check:
ffprobe -v error -select_streams v:0 \
-show_entries stream=color_transfer,pix_fmt -of default=nw=1 GX010001.MP4
arib-std-b67 is HLG, already HDR. bt709 with yuv420p is 8-bit SDR.
What action footage gains
Action cameras spend a lot of time pointed at exactly the content HDR helps with: sky, water, snow, and the sun somewhere in frame. Those are the surfaces that flatten against SDR's diffuse-white ceiling, and they are where the extra headroom is most visible.
They also spend time on high-contrast scenes with deep shadow, and there the constraint is the sensor rather than the encoding. Shadow detail the camera never recorded does not return.
The clip length limit
GoPro sessions run long, and conversion here takes clips up to 60 seconds. For a longer session, cut the segment you actually want first. Related: converting drone footage to HDR covers the same aerial and wide-sky material in more depth.
Questions people ask
Is Protune flat the same as log?
It is the same idea at lower strength: a reduced-contrast SDR curve that preserves grading latitude, not an HDR format and not a true log curve. Like log, it is meant to be graded, and like log it is not HDR no matter how flat it looks.
Does stabilisation affect the conversion?
Only through re-encoding. In-camera stabilisation is baked in at capture and irrelevant here. Applying stabilisation in post after converting will re-encode the file and strip the HDR metadata carried in the HEVC stream, so do it before.