A big part of the aesthetics is that the specific glyphs for parentheses are curved in a way that suggests enclosure.

Suppose we replace ( with ^ and ) with $. The visual aesthetics is ruined. Or suppose we swap them: ) means ( and vice versa. Ditto.

Let's try it:

  (let ((s nil))
    (dotimes (i 16)
      (setq s (rbt-insert (1+ i) s)))
    (pp s))

^ is opening, $ is closing:

  ^let ^^s nil$$
    ^dotimes ^i 16$
      ^setq s ^rbt-insert ^1+ i$ s$$$
    ^pp s$$

) is opening, ( is closing:

  )let ))s nil((
    )dotimes )i 16(
      )setq s )rbt-insert )1+ i( s(((
    )pp s((

the reversal is slightly better than using random glyphs, because we can pretend that ) ( is suggesting closure in a different way. I don't think I could get used to this no matter how much time I put into it.

I can't escape the conclusion that the shape of the glyphs is important. The parens, being shaped the way they are, are making the code readable.