I expected to be able to choose a date or time in the future
Nothing. You can't change the date or time of publication
Firefox 57.0/ macOS Sierra 10.12.6
Andrew Spittle also reproduced in Chrome (said it worked for him sometimes but not others).
826676-zd-woothemes
It looks like to reproduce this you have to try and set the date first before adding any content.
Just reproduced this error on my end as well, on Chrome 64.0.3282.119.
1747098-hc is reporting this as an issue even with content already added, across Ffx, Edge and Chrome. snag.gy screenshot from user in transcript.
Reproduced on own site on Win10/Chrome without content, could not reproduce with content on page, even just a title or single image.
@Automattic/tanooki will look into this.
[user] is reporting this as an issue even with content already added
This is likely because the user added content and then changed the date before the first draft save completed, which is the point when the editor receives a post ID. Currently, the calendar won't show a date until we get that post ID.
https://github.com/Automattic/wp-calypso/pull/23094 should fix that situation. If there is more follow-up needed afterwards, please feel free to reopen this issue.
I'm reopening this because it's still a bit broken for backdated posts and pages.

@Automattic/tanooki let's take another look :(
Verified that this does not function properly for backdated posts/pages.
@Automattic/tanooki is looking into this in the next few days
A few details that I noticed from a brief investigation...
post.modified is set to post.date instead of current timestamp when post.date is set and then draft is autosaved the first time
isBackDated returns false when post.date is set in past but post.modified is not set at all
@jsnajdr You might want to look into this one while in the area looking at other editor/scheduling issues.