What's actually going on
Your toilet doesn't flush by dumping water. It flushes by creating a siphon, and that siphon needs air to enter the drain system behind the water. That air comes from your plumbing vent, the pipe that runs up through your roof.
When wind changes how air moves through that vent, the siphon weakens. That's why the flush strength tracks the weather instead of the toilet's condition.
So if everything's fine on calm days and flushes go soft when it's blowing outside, you're looking at a vent problem in most cases.
The good news: this rarely means a broken toilet. The catch: the fix usually lives on your roof, which is why we don't call it DIY-friendly.
The likely causes, ranked
Start with a blocked vent. It's the most common by a wide margin. Leaves in fall, nests in spring, ice in winter, all of it narrows the opening. A narrowed vent flushes fine when air is calm but chokes when wind adds pressure swings.
Next is a vent that's cut too short or terminated in a bad spot. During reroofs, Techs have seen vents trimmed flush with the shingles because someone didn't want it sticking up. Code wants it up at least 6 inches, often more, and away from valleys and walls where wind swirls.
Third is a straight downdraft. Some vents just sit in the wind's path or near a taller structure that funnels gusts straight down the pipe. When that happens, the wind fights your flush directly.
In cold regions, add ice to the list. Warm air rising through the vent condenses and freezes at the cold top, choking the opening down to almost nothing.
Last, and least likely, the toilet itself is marginal and wind is a coincidence. Low water level or clogged rim jets can put a flush on the edge, where any little thing tips it into weak.