You can get a thin smartwool or polypro base layer from a ski shop for like $50, even cheaper on sale but it’s the wrong time of year for that. That’s probably all you would need unless you’re planning on running in a blizzard.
When I was in the military a bazillion years ago, our winter running uniform was a beanie to add to our t-shirt and shorts. That was it. If it was SUPER cold (like, below freezing) we’d get to wear sweatpants and a sweatshirt. Then we’d go run 4 miles in the dark. If you got too cold you just had to run faster. :)
Wool is fantastic for cold weather activity because unlike cotton it still insulates when wet.
I still wouldn’t want to RUN in wool, but it makes up some indispensable parts of my ski kit. And it has been tested!