You know, all this time, I had thought that you were just not making the proper routes to make getting out of the RED spawn less of an ordeal, but after looking at the map again and traveling straight up with noclip, I realized that it has nothing to do with the routes; it's a problem inherent in the map itself.
The spawn is literally at the back of the map.
As Sonic Sez,
That's no good.
This is what I suggest you do; you can choose to listen to me or not, but this slight overhaul would completely solve the issue you're having with people taking the low ground during the courtyard battle.
First and foremost, convert this into a spawn room. Close off the vent and the stairway up to the final cap. This is the biggest step. I know you don't want to put your work on that entire back area to waste, but trust me, I thought of what to do with that; let me continue.
When people spawn (this is before the penultimate point above the slope is capped, mind you), have them facing this direction. This is a great passageway that not only leads RED straight up to the high ground, but also gives them many vantage points, such as the Sniper deck that nobody uses.
This is a great location, but it's never used. By having the spawn be close to it, players, namely Snipers, will naturally gravitate towards it, while other players have either this stairway to go down, or the back area to the left of the map (which is also never used). Having the Snipers be out of reach would be problematic, but thankfully you have that area accessible to BLU with the stairwell going up into it, so a sneaky Spy could take down any troublesome Sniper in there and never be spotted. Good balance at work.
But there's one big issue with that room that stops it from being perfect.
This hole.
Why was it ever there? It serves no purpose other than to frustrate players. Especially if you convert the room below into a spawn room, that would make this hole even more useless. Remove the hole, make it a firm grating that nobody can slip through, and have that room down there just be closed off. Simplicity is good; the less unused rooms lying around, the better.
In order to adjust to having that area be a spawn room, make this doorway a spawn door. Like I said, having the spawn spit RED out nice and close to the battlefield, yet still far enough from the initial cart location that BLU has more than enough of a chance to push forward, is satisfying to both teams. However, when BLU caps the penultimate point, lock this door, and have an impassable door blocking off the stairwell from the spawn, so as to not confuse players.
Next, convert this open doorway into a spawn door, which will properly seal off this room as a spawn room. The design already perfectly fits having a door there; you've already done half the work. This door will always be accessible for the last two caps; on the penultimate cap, it might be wiser to take this doorway to flank the cart with someone else from above. However, it won't come into play as much until the final cap.
And as we get to the final point, you're probably still saying "Now hold up, I want people to be able to get to the high ground on the final point! I built that for a reason!"
And herein lies the solution.
Take this vent and convert it into a regular passageway. I know you want to be unique, but save the vents for the upper areas that actually have drop-downs; just crawling through the vents to get out of the spawn is a pain. Have a locked door here until the penultimate point is capped, so people don't take it by mistake; it's only there for the final cap. Now instead of making a path out of the entire vent network, though, this path will just lead to one spot.
Your stairwell leading up to the high ground at the final cap.
I bet you forgot that the vent system went here; we sure did.
So when the penultimate point is capped, change the spawn so that players are facing either the doorway on the lower end, or the hallway that leads to the stairwell. Passageways are no longer long and confusing, and are instead quick and simple to traverse.
In order to make the best out of the high ground at the final point, though, you might want to extend the cap a little deeper along the path, so BLU is exposed to RED fire longer.This will force BLU to have to travel through the building above the track (which, again, no one ever goes through) in order to dispense with the RED forces themselves, and, once they do so, they can jump right down to the cart. It would benefit both sides.
By making all of these small changes, the result will be a major overhaul of the map design, but for the better. Movement of players will be relocated to the upper areas, except for when the cart is in the lower areas. This is the answer to the frustration that the final spawn has always had.