The TV is not creating the same level of brightness/saturation/color on sdr content when that is displayed with hdr as it most likely supports only a low level hdr on sdr content (mostly happens on lcd/led screens, oleds are far better to get a good sdr on hdr picture).
So the problem with the washed out colors is normal and might be lessend by adjusting stuff like brightness when it is displayed (but would also affect hdr as you change it in that mode).
You might search the net if people might have posted settings guides for that TV which you might like to try.
BTW. Switching the color range to full was needed as hdr formats like dolby vision only work with the enhanced color space.
Not sure what you are trying to say here. SDR is SDR - sure you can chose to process it and push it into the HDR range - and many people do. This will make the picture look bright - but it will then mean an SDR signal will be displayed far brighter than the SDR portion of an HDR10/PQ signal.
PQ ST.2084 HDR (i.e. HDR10) is based on an 'absolute' HDR standard that maps specific video levels to absolute pixel light output - so a video level that deliever 100 nits is 100 nits on any ST.2084 PQ (aka HDR10) display unless that display is not following the ST.2084 PQ curve and is being overridden. Displays differ in their max brightness - and in those casese the PQ curve is tone-mapped to mitigate the max brightness limitations, and obviously there are better and worse performers at black level.
However 100 nits is a 100 nits (and is represented by 51.9% of full-range in a PQ ST.2084 signal) - any display correctly displaying the SDR portion of an HDR signal will deliver this 51.9% signal as 100 nits light level whatever the screen tech is if it is correctly meeting the standard PQ specs.
Some TVs will use processing that means they no longer track the PQ/ST.2084 curve correctly to let you make the picture in HDR brighter or darker - some in their 'Cinema' or 'Movie' modes won't.
I recently switched from a Sony FALD LCD to an LG C3 OLED with both correctly calibrated and the subjective performance was similar for both SDR and HDR content in picture brightness terms watching the same content - though the OLED clearly did better with blacks and there was no FALD backlight booming, and the OLED overall delivers a nice clean picture.
This is a really good primer on HDR PQ and HLG and SDR and explains a lot about why HDR can sometimes appear 'dark'. It's written by a former colleague who is an expert on both display calibration and motion picture colour science.
AIUI HDR10 may use 1-1023 rather than 64-940 (aka 16-235 in 8-bit) level space - which is why leaving a TV on Automatic rather than Full or Limited may make the most sense.