I don't know how this could be made to work, so I'll bow to your own thoughts on the matter, but it would look something like this:
There would be an "Invite" tab. You click on it, and you get a list of players (maybe subset by leagues/season, dunno).
You select whichever players you want to send an invite out to. This invite can be freeform text ("game at my place on the 12th" etc etc
They each get an email which they can click on to take them to a webpage (can even use a Google Sheets doc for this), and they can choose "Yes, Probably, Maybe, No" or whatever.
The responses can be saved just like any other tourney stat.
The TD software could track the statuses of the invites and send reminders to everyone who hasn't responded a week before the game, or send confirmation that the game is going ahead when X people have said yes etc etc.
TBH the more I look at this the more I can understand why you'd want to leave it well alone.