I carry 2 chains with choker hooks, each is 5400 lbs (and tow straps, and a snatch strap). The choker hooks are great, gives you adjustability capability.
I can run them in parallel if I want the heavier load.
But most importantly, it gives me a lot of flexibility, for example if winching something with the hi-lift (you got to secure the line before removing the hilift for the next "pull", I can alternate the chains).
And yes, chains are great to avoid damage to straps.
The ones I used are lower grade than 120. They are also 3/8" but only rated at 5400 lbs. About $15 for 15 ft each on sale at HF. The drawback of the low price is double the weight basically.