Published path of alternate output formats

No, there isn’t. I don’t much care for the way it works either.

If it were me, I would configure like this:

# uglyURLs = true

# [outputFormats.html]
# noUgly = true

# [outputFormats.json]
# path = 'output'

[outputs]
page = ['html','json']
section = ['html','json']

And then I would configure URL rewrites (redirects) on the server. Hopefully you have some control of that.