Here is my proposal:

Players could set non-binding times that are usually workable for them. For example as I live in Scotland and usually have no issue playing between 6-9pm on weekdays that translates to 1300 to 1600 Server time so any offer in that range is normally accepted but my opponents might not know that. Probably the easiest interface would be to have options for each day and then have a start and finish time.

Then when you go to the game forum it would get a post from TL bot with the following:

"Player A faces Player B in round X of the X Section

Player A has indicated on a non-binding basis they're usually free Monday 1300-1600, Tuesday 1300-1600 etc
Player B has indicated on a non-binding basis they're usually free Monday 0600-0900, Tuesday 1600-1900 etc

Please only set the time once you have received confirmation from your opponent and good luck!"

If player doesn't specify then it's left blank or replaced with a link on how to set a preferred playing time(s)

So once I see the above I would see what times usually work for the opponent and be able to offer something they'll accept and if the times are completely different then it means I know it will be hard to schedule and we need to see what can possibly work. In the random example above the time I'd offer is Tuesday 1600. It should be non-binding as players can occasionally have appointments etc that mean they're unavailable that week.

The main benefits are that it can probably save a day or two of negotiating finding out times that work for our opponent and also mean that if you have multiple games to arrange then you can make much more targeted offers as instead of offering a whole day you can offer the hour or two they say suits them best