

Perhaps it’s Luther, Line of Duty, Broadchurch, Endeavor, The Tunnel?
Or Midsummer Murder, Case Histories, Professor T?
It doesn’t sound like Death in Paradise, Brokenwood Mysteries, The Frankenstein Chronicles, Annika or The Fall.
BritBox has a list of tv series similar to Sherlock, Alienist, etc.
There’s Haden Clark but that’s gruesome. There’s a TV series about it on AppleTV and somewhere else I can’t remember but same title.
I’ll come back if I think of anything else. Good luck!
Ditto on used flatscreen (not-smart tv) from thrift shops; they’re light enough that you can take it with you when you move.
$500-$1000 is a good budget so you can buy new (non-smart tv) at the big box stores.
Yes, you should consider a good projector and a white bed sheet from the same thrift shop and iron it (I speak from experience). If the acoustics in your room is good then you may not need to spend a lot of money on speakers/headphones.
My friend couldn’t find black bed sheets so she bought the ugly stained ones and dyed them black and put them behind the curtains. We were able to do this setup in her backyard for the kids, too.
I also used a pre-primed canvas with my easel and worked just fine as soon as I got the angle right. Both came from JoAnn Fabric when they were closing the stores (maybe $35?). However, my Bose speakers and Sennheiser headphones were not cheap. Living in concrete and 2 different types of insulated walls are probably the reason why the acoustics suck.
That setup was so portable and worked really well that when my friend borrowed it she told me I wasn’t getting it back. You need to decide what you want to spend top dollar on: VR headset, projector, TV, etc. I’m actually curious how you end up with your entertainment setup. Good luck.
EDIT: I gave my old Harman Kardon speakers and subwoofer (from my old PC) to my friend and it still sounded good.