The tricky part with turmeric is bioavailability - curcumin barely absorbs on its own, which is why most supplements add black pepper extract (piperine) to help your body actually use it. Without that, you're basically flushing money down the drain. Your grandmother's approach of mixing it in warm milk with fat is actually smarter than a lot of capsules because fat helps absorption, so she might be getting more benefit than people realize.
The research shows curcumin does have real anti-inflammatory effects in lab settings, but the human studies are all over the place. Some people see measurable relief, others don't notice anything. A lot depends on your genetics, what's causing the inflammation, and whether you're actually getting a dose that's bioavailable. For gym-related joint pain specifically, I'd try it for a solid month or two - if it's not helping by then, it probably won't. The thing is, it's not expensive enough to wreck your budget if it doesn't work, so the risk-reward isn't terrible.
What might actually matter more than the turmeric itself is whether you're addressing the root issue - overtraining, form problems, recovery time. I've seen people chase supplements for pain that went away once they just backed off a bit or fixed their technique. But if you're already doing that and want to hedge your bets, the milk-with-turmeric routine costs almost nothing and isn't going to hurt.