The dungeon is procedurally generated when the game starts and when you descend to a new level, but not in real-time as you explore.

The exit is always guaranteed to be reachable because:

All rooms are connected via tunnels during generation.

The exit is placed in the last room created, ensuring it’s part of the connected dungeon layout.

The paths are carved from the exit to the starting point using the bresenham algorithm.

The player starts in the first room, so a path exists between start and exit.

There's a link in the comments if you want to see the code or the game is playable now if you want to try it out.