I started riding fixed gears back in the early 70's as winter training
That takes me back. I remember you early-'70s bike-boom guys. When I started racing on a track bike in 1964, there were only 1,500 bike racers with ABLA licenses. By 1973 or so, suddenly there were guys showing up for races who I didn't recognize!
To the OP, although I love my fixed gear bikes (1968 Peugeot Professional w/Campy Record, aluminum Specialized Langster from the first year of production) and still do very hilly rides of up to 6 or 7 hours on them, don't know whether I'd recommend starting out with a fixed gear for your first single-speed bike. But if you have a relatively car- and hill-free area to get used to it, it's worth considering. There's nothing quite the experience of riding a fixed-gear bike.