Your framerate is always about what your game's processing is bound by.
Most games have been so lightly using IO - because it's been so slow - so it's rarely been the bound to date. You load a lot upfront into memory and then stream a very small amount at runtime. However, stutters in framerate at runtime due to temporary intensive IO actions aren't so unusual.
Going forward, if games do start leaning in on relatively intensive IO use at runtime, the SSD could become more important to framerate. There are a couple of examples out there currently, where if you run the game on a standard HDD, the performance is terrible. You become IO bound on a regular HDD.
If you take a game that needs - I dunno - 1GB/s of streaming/runtime bandwidth from storage, and try to run it on a slower HDD, your framerate will suffer, yes - so in that context a SSD becomes key to good framerate.
To date very few games lean on storage IO like this. But that might start changing.
Side note: Sony or MS or whomever, should do a tech demo where they craft a very IO intensive scenario, run it on a PS5, then run it on a PS5 with a slower SSD, and show the difference. Do the Spiderman 'fast-as-a-jet' demo, and show what happens to the framerate on a slower drive. Indeed, if they're so confident their SSD is faster than anything else, they could do the same demo on a super powerful PC with a standard SSD, and demonstrate a difference. If the bound is in storage IO, it doesn't matter how much faster the PC's cpu or gpu would be... it would really hammer home the point, or the benefit of their approach.