Sugar beet is useful for bulking up and, soaked, for moistening feeds. It's not as nutritious as ewe nuts or coarse mix.
We use it to moisten the grass pellets we feed, otherwise our greedy sheep wolf the pellets so fast they make themselves sneeze and cough.
. The house cow gets it to moisten the mollichaf we use to bulk up her milking-time feed.
On the moorland farm, where we were feeding 450 Swaledale on the moor, the previous incumbent had used a mix of sugar beet and ewe nuts for the first four weeks of feeding pre-lambing, then all ewe nuts the final four weeks. He said he did that so that they'd be on a 'rising plane of nutrition' for lambing, but that as the price of sugar beet had risen and it was no longer significantly cheaper than the ewe nuts, we may as well just feed a bit less of the ewe nuts for the first four weeks.
So no, it's not something you 'should' be feeding your Castlemilks. If you wanted to give them a bit more quantity of feed but not too much more protein, mixing in some sugar beet (either dry pellets or soaked shreds) would be one way to achieve that. Feeding s lower protein feed would be another.