Unfortunately there aren't a lot of well planned clinical trials w joint supplements that use a double-blinded study design (control and treatment are randomized amongst enrolled animals and owner and caregiver don't know what group they are in) and non-biased outcome measures. That's the study design that is needed to prove efficacy.
Without that you are left with something called caregiver effect/bias. Mike Conzemius' group described this in small animal orthopedics. It's basically the placebo effect for pet owners. Just because there is a perception of improvement doesn't mean there is actual improvement. And vets are just as susceptible to it as clients are. We use this often in our teaching rounds with the veterinary students.
https://pubmed.ncbi.nlm.nih.gov/23113523/OP, you would need to look for a study that showed supplement x is superior to placebo using that study design to find the answer you are looking for. Unfortunately those studies are very very sparse and companies don't want to fund them because they often show no improvement over control, and that won't sell product. It's the unpleasant reality of joint supplements in small animal medicine and humans too, but I'll stay in my lane. They are not FDA regulated as they are considered neutraceuticals.