Aside from cost, is there any disadvantage to using thicker wire than required?
Electrically, thicker is better in this case.  However, thicker cable has other disadvantages beyond just the extra cost:

Less flexible.  This makes it harder to install and harder to work with in general.

Bigger, which means it might not fit thru some small holes or tight places.

You get diminishing returns with AC due to the skin effect.  Thicker is never worse than thinner electrically, but after some diameter the extra area gives you less and less return.  This effect is proportional to frequency, so thicker cable is more useful for something like 60 Hz power as apposed to 10 kHz signal to a loudspeaker.