I only recently read about the garlic and ginger being detected as the main source of probiotic lactobacillus bacteria, rather than the cabbage, it was from a 2015 Korean study into Kimchi. LAB was found in 5/5 garlic samples, 3/5 ginger samples, 2/5 leek samples and only 1/5 cabbage samples, of which it was weak. So ability to act as a starter culture for LAB seems to correlate with the strength of the prebiotic, so I imagine other strong prebiotics like chicory root and Jerusalem artichoke might also work.
https://www.ncbi.nlm.nih.gov/pubmed/26133985
I haven’t had an issue with slimyness with carrots, but I did have an issue with a beetroot sauerkraut I made in June-July, I think it was simply too hot, and I hadn’t used the coolbox method. How much salt are you using? Are you thoroughly squishing it all up and covering with a cabbage leaf and weight so the brine covers the whole mixture?