why air becomes “wobbly” (and, thus, kind of visible) when it’s hot
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It is not because the air is hot it is because it is warmer than the surrounding air.
The first part is warm air is less dense the cooler air so it rises. It will not rise in a perfectly smooth, there is always wind and turbulence, Look at how frames in an outdoor fire behave.
The second part is the index of refraction of warm air is different from cooler air. If light passes between medium with different indexes of refraction its direction changes.
This is why if you put a straight stick into the water and look at it from above it does not look straight. This is also how lenses work.
The difference in the index of refraction is less between cold and warm air compared to air and glass or water so the amount the light gets bent is lower.
Home muck the light bend depends on the different index of refraction and the angle between the light and the surface between the two materials.
The rising warmer air will not have a smooth surface or really a surface at all it will be a gradient but let’s ignore that. Think instead of rising warm air as something similar to water where the surface moves amount but vertically. Then let’s look at the reflected sky instead of what is below the water it is often simpler to see. If you have water where the surface moves the reflected sky will be wobbly, If you are in a pool and look out through the surface what you see is wobbly.
The effect will be more pronounced for water than how air but the phenomena is the same. the light passes through material with different indices of reaction that change the shape all the time and light will be bent in a nonuniform way ie it wobbles.
You can see the sky reflected in the air too like the water. You need to look with a low angle relative to the “surface” of the hot air and there has to bee a high-temperature difference. You get that if you for example look along a hot road like https://qph.cf2.quoracdn.net/main-qimg-68be869c597075a0f14feb78512048ae-lq
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