My understanding and experience are that this viscosity happens when the roots are very sweet and that there is some alcohol or yeast production during fermentation because of the extra sugar in the roots. My last batch of fall beets did not produce this thick brine, but it’s common, and I know many people seek it out as “proof” that the beneficial bacteria is in there.
I don’t shred my carrots and beets. I cube them or cut them into thicker julienne slices. With the increased surface area you’re creating with the shredding, I’m not surprised that you’re having this texture issue. However, you will need to use brine if you cut the beets into larger pieces.
Another option would be to combine the shredded beets with a shredded brassica? Like a turnip or part of a cabbage? You might avoid having to add extra brine this way, and you might like the flavor. I really love beets and turnips together.
The good aspect to all of this is that you must have an awesome, fresh source for your vegetables. To have a root hold that much sugar at this time of the year is a wonderful sign of plant health.
All best to you.
-cathy in Oregon