Integrity is a balanced formula to complement the forage portion of your horse’s diet. The formula is balanced for the 48+ nutrients required by the horse including biotin, zinc, copper, calcium, etc. which is a few of the ingredients in the over-the-counter supplements that you specifically referenced. The Integrity formulas also contains soluble fiber sources which are in the”family” of prebiotics as well as it contains a yeast culture. I do not generally recommend over-the-counter supplements for horses that are being fed a balance formula to complement the forage portion of the diet. If a horse is maintenance fed and can be maintained on only hay or pasture, then depending on several factors including geographic location and forage sources I may recommend a generic-complete vitamin/mineral supplement.
The diarrhea is most likely not related to feeding. Considerations to explore include deworming schedule, dental health, time of day worked, temperature where stabled and temperature when worked, horse’s temperament prior to work, and overall body condition relative to exercise deportment. I am not a big fan of daily psyllium supplementation and there is a fact sheet on those thoughts in Dr. Bray’s Corner. If your horse has been diagnosed by a DVM with ulcers and joint problems then I would visit with your vet about more proven treatments then the over-the-counter supplements. The Integrity formulas provide adequate levels of biotin, lysine and methionine which are the primary ingredients being marketed by your hoof supplement.