I just tested this on FCP 11.2 and Tahoe 20.1 on an M1 Ultra Mac Studio and an Apple Studio Display with the default P3-600 preset, also the BT.709/BT.1886 reference preset.
Using an Rec.709 H.264 clip in a Rec.709 timeline and an SDR library, Save Current Frame to .jpg looked just the same in Quick Look and in FCP as the original video clip in FCP. I also tested it on an Eizo CG2700X monitor connected via USB-C (AltDisplayPort) using the standard Eizo ICC display profile. They looked the same on that monitor.
I also tested it using FCP 11.2 and Sequoia 15.7.2 on an M1 Max MacBook Pro 16. Save Current Frame to jpg also looked the same in Quick Look and FCP as the original Rec.709 video clip.
Is your FCP library SDR or Wide Gamut HDR? Is the project timeline Rec.709 or something else?
In Finder, select your exported current frame and do CMD+I. What does that say for Color Space and Color Profile?
For your video clip from which you did Save Current Frame, play that in Quicktime Player and do CMD+I, open the Video Details section, and tell us what that says, especially for the three-digit code points. What kind of camera was that shot on and what codec is that?
We also need to know your display color profile or display LUT. That is shown in macOS System Settings>Displays>Color Profile, or for newer MacBook Pros with an XDR display or Apple Studio Display it will be a reference preset. What does that show on your machine?
There was previously some kind of issue in this area, but I thought it only happened with a Wide Gamut HDR library, and it was a MacOS issue not caused by FCP. I can't clearly remember.
That said, I don't think it's guaranteed that a save current frame from FCP or a screen cap will show the same color behavior as the original video clip. Those are really not the ideal methods to save a color reference from a video. The best way is to use the FCP range select tool, select a single frame on the timeline and export that as a single-frame *video* clip. That will have all the same color metadata as the original clip. You can select a single video frame by pressing I to select the "in" point, hold down K and press L to go forward one frame, then the letter O to set the "out" point. Share that from FCP and it will export a single-frame video.
A Quicktime video file contains NCLC tags that ColorSync uses to color match to the display profile. There is no equivalent to NCLC color tags in a .jpg or .png still. That said it seemed to work on my two test machines using FCP 11.2, Tahoe 26.1 and Sequoia.