I know others have made a post about their bother with the current situation but I'd like to make this post as a suggest as to what the dev team could potentially implement to solve this issue.
When you click on "Proceed" for instance solo content, rather than start the instance and CS, you queue it up for a Duty. By doing this you allow the server to reserve spots for players waiting to enter. When an instance is open the queue will let the player know and the Cutscene will proceed to play followed by the instance.
By doing the above I feel you will not only relieve stress on the server but make it a lot easier for player to get through these solo instances.
If anyone else thinks this is a decent idea I'd ask if you could upvote this and give it some attention because maybe they can actually implement it into an emergency maintenance to help with the situation.
And a reason why I do believe this is plausible is because this is something that is already done with Chocobo racing, you don't queue up for the instance through Duty Finder but from talking and confirming with the NPC so why not do the same for solo instances.