Most Xbox 360 games of the first years worked without any HDD. A few later games needed a small HDD for some caching and for patches, but most of the data still was loaded directly from the DVD.
Most PS3 games only needed a small part of the HDD for some caching (usually as mandatory installs of tiny fractions of the game) and some patches, but most of the data still was loaded directly from the Blu-ray.
Xbox 360 Core and Xbox 360 Arcade existed, the PS3 Superslim with 12 GB existed.
You needed to buy a HDD for the arcade for certain games to run and you can't install all acts of MGS4 on a PS3 Super slim.
The fact that gimped consoles existed didn't mean games were held back to be able to work on those!
But of course not all games will need the speed of an SSD. Some will, if they won't be held back. ps3 and 360 had other limitations as well making the slow HDD and Blu-ray speed less of a problem. 2 generations later, HDD is the biggest bottleneck.
Just a little random tidbit of information: PS2 Texture buffer bandwidth: 9.6 GB/s, almost the same as the max throughput of the PS5 SSD. Pretty amazing.