Yes, this is all too easy to do, particularly with so many various pro- and prebiotics in foods these days. This is one of the reasons I took our cats off all their commercial raw foods last summer: sorting out what's where in what quantities would have been nigh on impossible, given the cats' menu!'playing with' or experimenting with a cat's gut bacteria can cause real problems.
Yes, the "doesn't have all the extra crap" was one of my big reasons for buying it! It has everything I want in a supplement (our cats definitely have inflammation issues) but none of the bad stuff. II second the recommendation for Adored Beast Gut Soothe. It’s fantastic and doesn’t have all of the extra crap that tends to be found in things like chews. If you’re seeking out probiotics due to concern about your cats microbiome, there’s always the Kitty Biome test by AnimalBiome too.