The issue of how to handle B&B plants is a bit controversial in the industry. IME the majority of landscaping companies do more-or-less what you are describing and usually it works out fine, at least in the short run. In my case I make sure as much of the cage and burlap is removed as possible, usually 70% - 100%*. Over the long term the cage may create problems with root development but this probably wouldn't show up for a decade. If the burlap was partially synthetic or treated with a extra strong dose of fungicide it is more likely to be part of the problem in the near term. I doubt however, that these rather dubious practices killed your tree unless the contractor left exposed burlap on the surface where it can act as a wick to pull moisture away from the root ball.
If the tree was healthy when it arrived (not a given, conifers can be almost dead before showing obvious signs), if the contractors didn't let the ball completely dry out before planting, if there was not a major soil discontinuity between the ball and site soil (e.g. a high clay content root ball being planted into a high sand content site), if there were not large air voids left when planting and if the Thuya was watered in correctly at planting it probably wasn't actions of the contractor that killed it. That is, however, a long list of ifs. Under watering or over watering by the homeowner are also not infrequent causes of plant decline or death. Anything anyone can offer at this remove is no more than a guess.
Arborvitae are over promoted as "shade tolerant". They are, to a limited degree, but when grown in anything more than light shade they tend to develop a very open habit which I find unattractive. At least in the northern tier states I wouldn't plant a Thuya in situations where it received less than five hours per day of full sun or dappled sun for eight hours or more.
*Some nominal experts recommend removing not only the cage and burlap but all of the soil from the root ball. I've never met anyone who actually uses this practice on a regular basis but I have no evidence that it doesn't work if done carefully. This would dramatically raise planting costs.