I've ridden fixed for 5 years or but currently ride single speed, or , if steep climbing is involved, geared bikes.
The advantages of fixed are, at least for me:
It requires that I pay attention, which after riding the same routes for 25 years, is good as I tend to get high on endorphins and space out.
Climbing is easier in a given gear than the same gear in SS or geared bikes and better traction control on slick or icy roads.
Fixed is the simplest mechanically , no freewheel to wear out, no derailleur adjustments and maybe no rear brake either. More than any other bike I felt like I was wearing the bike rather than riding it
.The disadvantages are only one gear choice, spinning too fast on downhill stretches and a little awkardness in stopping invterms of pedal position especially with SPDs.
The advantages of SS over fixed are largely going down long grades and being able to rotate the pedals without moving the bike, as when stopped. Maybe not quite the "one with the bike feeling" but close.
I have a number of bikes I ride frequently and I ride my Surly Steamroller set up as SS often. I find it fun and comfortable. I could only have 1 bike it would be my Brompton, but the Surly is a close second