Yes, yes and a million times yes.
What we've seen over the last 18 months or so is a trend from storage drive speed being almost irrelevant in terms of game performance (aside from loading-times), towards it becoming one of the most critical factors. I'm guessing that, as you suggest, it is caused by a combination of the move towards more "open world" games, and an increase in the detail-level, and hence size, of game-assets.
The first game I'm aware of where it was a serious issue was Watch_Dogs. You'll recall that the PC version of that game took quite the hammering at launch. Not just because the game was rubbish (although it was), but because a lot of players were experiencing severe in-game stutter. In fact, I noticed it myself when I first installed the game. While it was ok on the indoor on-foot sequences, the moment I started moving around the open-world city, particularly in a car, it was stuttering constantly. Then I noticed that the timing of the stutter was perfectly in-sync with the activity of my disk-access light. So I reinstalled the game on my solid state drive and, surprise surprise, the stutter was completely eliminated.
There have been a large number of games since then which have been affected to varying degrees. Particular culprits include:
Dragon Age: Inquistion (infrequent but very severe periods of stutter when moving across invisible transition-points when running from a mechanical drive).
Far Cry 4 (at the lower end of severity, but stutter noticeable when opening doors or entering vehicles while running from a mechanical drive).
Borderlands: The Pre-Sequel (mild stutter and long texture-loading delays when running from a mechanical drive - for some reason much more noticeable than in Borderlands 2, which used the same engine).
The Witcher 3 (mild but noticeable stutter when moving at speed on horseback, particularly around towns/villages, when running from a mechanical drive).
Pillars of Eternity (yes, even in a top-down 2D RPG, there's stutter after issuing some commands, particularly for spells whose visual effects need to be loaded, when running from a mechanical drive).
Batman: Arkham Knight (the original unpatched release essentially unplayable due to severe stutter when running from a mechanical drive).
I suspect that poor optimisation is also a factor in some of these cases. After all, the console versions mostly avoid this stutter, despite the fact that they contain fairly slow and crusty mechanical drives. Irritatingly, some of the performance-comparison sites out there, particularly the (formerly excellent) Eurogamer Digital Foundry don't do drive speed comparisons and seem to use SSDs by default, so they don't pick up these issues.
There are still a few major releases that appear to run well from mechanical drives on PC; Shadows of Mordor and Metal Gear Solid 5, despite being open-world games, don't seem to have particular stuttering issues. But we're getting to the point now where if you want a decent experience in PC games, having a solid state drive (and preferably a large one) is as important as having a decent graphics card.