Interesting question.
I agree, a lot of units in astronomy are a bit crazy, but I do find arcseconds are a very useful size. They are close in size to typical seeing conditions, and a small number can be used to describe the size of planets and many other celestial objects.
Arcseconds are also quite handy when dealing with distances measured by parallax, like by the Gaia mission, as they are used to define the distance in parsecs. The distance in parsecs is defined as one divided by the parallax in arcseconds.
Cheers,
Andy