I carry link chains, it's required in oregon in the winter, and occasionally they do check. And I have them for the trailer. Oregon requires "drag chains" on at least one trailer axle when chains are required. Now, I've only had to use chains once when towing, it was NOT a pleasant experience. All the other times, I just wait out the weather.
I will never use cable chains, way to many of cable chain pieces laying beside the road after a storm.
My experience in oregon is that by the time they actually have a check stop for chains, you (well I) will NOT be going any further.
But the I-5 summit border into CA can be a different story. Last winter our son going down to San Diego was required to stop just into CA and install chains even though the road was clear of snow. CA state police pulled traffic off the highway at an off ramp and would let anyone without chains back on. He put on chains followed a few trucks and when they pulled over to take of chains he did so as well.