Export and timeline colours are very washed out compared to original footage

Ann is correct. You need to use color management these days, as things are far more complex than they used to be.

Another issue that you are probably hitting is how you determine what Premiere is "doing". 

Using QuickTime Player on a Mac ... unless your Mac has Reference modes, and you set that to HDTV .... will show a wrong image. Period..

Because Apple for some unknown and irritating reason, does not apply the by-standards spec for display transforms for Rec.709 media playback. 

The spec? The long-used standard for all Rec.709 professional work is the full Rec.709/Bt.1886 set: sRGB primaries, D65 white point, and display transform roughly equal to gamma 2.4. All broadcast and streaming media is produced to that standard.

Macs without reference modes set to HDTV will use essentially gamma 1.96, which produces a screen image of lighter shadows, and less apparent saturation. But on those screens only!

ALL other screens ... Macs set to HDTV, PCs, most TVs and Android devices, and all systems compliant with professionl broadcast standards, will show a darker image of the same file.

To check this, use VLC or Potplayer on any non-Referece mode Mac. You will probably see the correct, darker image.

And yea, this is a pain.

Its all about color management.

Color management in Premiere Pro 

2025 Premiere Color Management V25 2 - YouTube