My wizard has access to dimension door and misty step. The problem is that in my campaign, the Astral Plane is broken, inaccessible, and mingled with the Material Plane, therefore unable to be traveled through. Do teleportation spells transport targets through the Astral Plane?
In D&D 3.5 this was true, as the quote below shows. In D&D 5e nothing in the rules makes it necessary to travel through the astral plane to teleport. Magic has changed. (Plenty of things with the same name work differently in different editions of this game).
A teleportation spell transports one or more creatures or objects a great distance. The most powerful of these spells can cross planar boundaries. Unlike summoning spells, the transportation is (unless otherwise noted) one-way and not dispellable. Teleportation is instantaneous travel through the Astral Plane. Anything that blocks astral travel also blocks teleportation.
The D&D 5e rules do not add the Astral element to this spell:
This spell instantly transports you and up to eight willing creatures of your choice that you can see within range, or a single object that you can see within range, to a destination you select. If you target an object, it must be able to fit entirely inside a 10-foot cube, and it can’t be held or carried by an unwilling creature. (Basic Rules, p. 102; PHB p. 281)
If you have decided to nerf those spells, you should probably let them refund those spells, as they wouldn't have picked them if useless.
Characters do not travel through the Astral Plane when teleporting.
From the PHB
You teleport yourself from your current location to any other spot within range. You arrive at exactly the spot desired.
RAW, the only relevant locations are where the spellcaster starts and where he ends up. Based on the duration of the spell (Instantaneous), there isn't time to be in another plane than the one you are in. You are just making a door to another point you can imagine within range.
Now of course, since the means is not mentioned, you could easily homebrew that the players travel through the Astral Plane when teleporting, but that is not how the spell is written in the PHB.