When we're looking at whether or not Yucca helps with joint pain, we need to look at the compounds in the plant itself, and whilst there is some indication that a few of these can have benefits, the volume of them is relatively low. Meaning that realistically this is case similar to pineapple, where extracts from the plant, or in this case root, could have some benefits, but you need to consume a very high volume. Meaning that a small amount of Yucca powder or as a asupplement isn't all too likely to do much.
Unfortunately, this means it's inclusion in products like Heal and Soothe isn't ideal, despite the supplements overall popularity.
As for the compounds themselves, this includes resveratrol, which has been shown to be effective for joint pain relief and is used in several similar supplements in it's pure form.
Along with resveratrol, Yucca Root contains a range of polyphenolic compounds known as stilbenes. These possess potent antioxidant properties, which can help protect the body against oxidative stress, otherwise known as the damage caused by free radicals.
Additionally, these compunds in Yucca act as inhibitors for the bodies inflammatory processes which can make joint pain worse. Notably NFkappaB, which stimulates the synthesis of inflammatory agents.
Modern Research Findings on Yucca for Arthritis
Scientific studies have corroborated the traditional use of Yuca Root for arthritis relief. Research indicates that the saponins and polyphenols found in the plant contribute to its anti-inflammatory and analgesic effects. By inhibiting the activity of inflammatory enzymes and reducing oxidative stress, Yuca Root may help alleviate joint pain, swelling, and stiffness associated with arthritis. Further studies are warranted to fully understand the mechanisms behind Yuca Root's potential benefits for arthritis sufferers. 
The catch is that you would likely need several grams of Yucca from the data currently available, but even then the extract would need to be standardized for a high yeild of resveratrol and saponins to have any effect. Resveratrol needs to be present in at least 100mg to have any effect for joint issues.  As we can see from the table below, you'd need 5 grams of yucca bark to yeild 100mg of resveratrol and we'd need even more from the plant (which also references the root in this case. 
In short the research on if Yucca can have some benefit for arthritis and joint pain seems reasonable, if it is consumed in a high enough volume. It is generally simpler to get the saponins or resveratrol isolated from other plants with the same compounds that have a higher yeild.
Other Benefits of Yucca
Vitamin C and White Blood Cells
One notable thing about Yuca Root is it's high vitamin C content, which plays a vital role in immune system function. Vitamin C stimulates the production and activity of white blood cells, which are essential for fighting off infections and viruses. Incorporating Yuca Root into your diet or taking it as a supplement can help boost your immune system's response to pathogens.
Yucca as Sunscreen
Exposure to the sun's harmful ultraviolet (UV) rays can lead to skin damage and increase the risk of conditions such as sunburn, premature aging, and skin cancer. Research has shown that Yuca Root exhibits photoprotective properties, potentially shielding the skin from UV-induced damage. In fact, some studies suggest that Yuca Root may offer better protection against the sun's harmful rays than certain commercial sunscreens. 
The extract can also be beneficial in addressing various skin conditions, including dandruff, balding, sores, cuts, and certain skin diseases or infections.
Yucca and Blood Sugar Regulation
There have also been some studies on diabetic rats that suggest Yucca could have some mild impact on glucose and insulin resistance. Although the effects we relatively mild and haven't been reproduced in human trials currently. 
Yucca Root Side Effects
When consumed in normal food amounts, Yuca Root is generally considered safe. However, taking Yuca Root supplements in high doses or for prolonged periods may lead to side effects such as upset stomach, bitter taste, nausea, and vomiting.
1 - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1440857/
2 - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9465647/
3 - https://www.researchgate.net/profile/Elena-Urso-2/publication/318112237_Quantitative_determination_of_Oleuropein_in_Organic_Olive_Oil_by_ESI-MSMS_and_Isotope_Dilution/links/595a4f0f0f7e9b897eab3649/Quantitative-determination-of-Oleuropein-in-Organic-Olive-Oil-by-ESI-MS-MS-and-Isotope-Dilution.pdf
4 - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8433717/
5 - https://pubmed.ncbi.nlm.nih.gov/18413208/