We were there a few weeks ago, 2 days at the unknown entrance--Bechler and two at Mammoth Hot Springs. Every cg we passed was closed but we only drove the West Yellowstone to Mammoth Hot Springs road and maybe 20 miles east on the north road. I never got a clear look at their cg status board so you should check online.
Crowds seemed to be normal, this is our 5th or 6th visit here. Visitor centers are closed. Mammoth HS traffic awful as usual.
We stayed at an excellent boondocking site 15.5 miles north of the post office at Gardiner off 89. We were doing a drive by mainly my wife wanted to see some critters.
The road to Bechler is ten miles of extreme washboard BTW but the Cave Falls CG is good and the falls and trails excellent.