I use this splitter with an EDID emulator plugged into port 1 set to 1080p/60 and my non-HDCP monitor on port 2.

https://www.amazon.com/dp/B08D9BCX1R?ref=ppx_pop_mob_ap_share

By now I have a more complicated setup that allows me to watch via my Fire TV stick on my computer, but what I would recommend you is to get a 4k Fire TV stick (ideally when it's on sale) plug it into your monitor's HDMI port (if it supports HDCP 2.2) and use the monitor audio passthrough to get the audio onto your headphones or speakers (assuming that your monitor has such a thing; if it doesn't, this won't work). If your monitor has no HDCP, but an audio passthrough, you can go for an HDMI splitter to strip it. But such a setup will cost you approximately 80-90$ (I paid around 300 for mine) which is neither affordable nor worth it to most people.

Yes everything you just said is what I think is happening. I am looking at this splitter which seems to advertise being capable of what I need: https://smile.amazon.com/gp/product/B08D9BCX1R/ref=ppx_yo_dt_b_asin_title_o00_s00?ie=UTF8&psc=1

If you need to pass up to 4K 60 hz you get one of these:

4K@60Hz HDMI Splitter 1x2... https://www.amazon.com/dp/B08D9BCX1R?ref=ppx_pop_mob_ap_share

If you have an Xbox X or Ps5 AND a 4K 120hz tv you get one of these:

8K@60Hz 4K@120Hz HDMI 2.1... https://www.amazon.com/dp/B0B5ZQK9BZ?ref=ppx_pop_mob_ap_share

You do NOT need to connect your hue sync to the TV at all. One side splitter goes to hue, other side goes to tv, for each device.