Faster storage is usually described in terms of loading times, which understates what changed. The constraint it removed had been shaping how games were built long before anyone measured it.
Mechanical drives punished scattered reads
A spinning disk reads by physically moving a head to a position and waiting for the platter to bring the data underneath it, which takes a comparatively long time.
Reading a large block laid out contiguously was reasonably quick, but reading many small pieces from different places multiplied that mechanical delay for every request.
Games therefore packed their assets into large ordered files and read them in long sequential runs, because the layout mattered more than the total amount of data.
Level design absorbed the limitation
If loading an area takes many seconds, the design has to hide that time, which is where corridors, lifts, crawl spaces and slow door animations came from.
Those moments look like pacing decisions and often are, but they exist primarily to give the storage system a window in which to fetch the next section.
Open worlds solved it differently, streaming terrain in low detail from a distance and improving it as the player approached, which is why distant scenery used to shift visibly.
Solid state storage removed the seek penalty
Flash memory has no moving parts, so the position of data on the drive has little bearing on how quickly it can be retrieved.
Small scattered reads become nearly as efficient as large sequential ones, which means assets no longer need to be duplicated across the disk to sit near where they are used.
That duplication had been inflating install sizes considerably, so removing it reduced the amount of storage a game needed even as textures grew larger.
The bottleneck moved rather than disappeared
Once data arrives quickly, the limiting step becomes decompressing it, since assets are stored compressed and the processor has to expand them before use.
Modern consoles include dedicated hardware for that decompression precisely because a general processor became the new constraint once the drive stopped being one.
On PC the same problem is handled by moving decompression work to the graphics card, which is why storage benchmarks alone predict loading performance poorly.
Design has only partly caught up
Games that assume fast storage can change location instantly, which enables structures built around rapid movement between distant places rather than around continuous space.
Most releases still support slower drives, so the design cannot depend on the assumption, and the benefit appears as shorter waits rather than as new structure.
The clearest changes show up where a platform can guarantee the hardware, which is why console generations produce sharper shifts in structure than PC releases do.