I chain my trailer because if I have to stop quickly I don't want my trailer trying to pass me.
This happened to a friend of mine on his first winter trip in the
Sierras. He and his family were lucky as they were not injured other than scrapes and bruises but his whole rig was totaled.
I lived in Northern Cal. for 5 years. When it snowed I had 3 or 4 feet on the ground at my home for months and the streets and roads were ice for days at a time.
Even though I had a 4X4 and did not need the chains to go on occasion I used chains on the rear wheels because chains increase your STOPPING ability considerably.
When you have chains on the TV your trailer is going to skid
when the brakes
lock up without chains and try to pass you.
My experience and advice is free so you can take it for what it's worth.