You could easily argue the games themselves are being pulled console-ward. And the games themselves are an important factor in console controllers. Console controllers not only shape the games that can be created for the console, the perceived needs of future games help shape the designs of the next controllers.
Looking at it from the console side, the progression of console game pads is pretty much just the natural evolution of the foundations set by the SNES controller. Sony reshaped it and added a couple more shoulder buttons, then added dual analog sticks to better suit 3D gaming. Analog triggers would continue to come and go for stuff like driving games. Pressable thumbsticks added a couple more buttons without adding separate buttons. Simple gyroscopes and accelerometers came out of the motion controller boom (though they’d been one-off gimmicks in earlier handheld games). Sony’s touch pad was a simplification of the touch screens of Nintendo and Sony handhelds.
You’ve missed a number of blips in that list of controllers. As well as the rise and fall of accessory controllers, which became less prevalent as the base controller’s functionality grew. A number of consoles would offer PC-like keyboards or mouse support, often just as it would a dedicated Mahjong controller or flight stick or steering wheel.
You’ve also missed how close some of those early consoles were to personal computers, some even were personal computers with some features shaved off. The Atari 5200 was an Atari 400 computer with features cut. The Atari XEGS was literally an Atari computer marketed as a game console, with the keyboard sold separately. Sega’s SG-1000 (a precursor of the Master System) was a cost-cut SC-3000 personal computer rushed to market to compete with the Nintendo Famicom (NES). Nintendo’s Famicom itself got its name from “Family Computer”; in Japan it had a keyboard accessory, a cartridge that allowed you to program BASIC, a modem, a cassette recorder for saving data, and eventually a floppy disc drive. (The Famicom Disc System even replaced carts in Japan, which in turn led to the addition of battery saves to Western NES carts.)
Of course early consoles had other schemes. Mattel apparently decided the future of game controllers was the touch-tone phone, as the Intellivision had a 12-button phone pad with four side “action” buttons and a “16-position control disc” instead of Atari’s 8-position joystick. The Magnavox Odyssey 2 launched in the late 70s with a full keyboard built into the console.
As for Sega’s Dreamcst controller…
Instead of improving on what others had done, Sega constantly lagged behind or misread what gaming wanted/needed. Despite releasing after the NES was already established, the Master System offered only two face buttons and no utility buttons. The Genesis controller was a clear evolution of the Master System pad, but functionally it only caught Sega up to what the NES pad had already offered five years (and a console generation) prior, even though the limitations of the NES button count was already showing. The fighting game boom forced Sega to release a 6-button controller mid-life, but Genesis games still had to be designed with 3-button play in mind.
Sega at first seemed to hit a winner with the Saturn pad, only for it to turn out to be the wrong path years later. The Saturn pad added two shoulder buttons to the Genesis pad, which would not only be functionally similar to what Sony would offer with the PS1, the six face button layout was superior for fighting games. While analog was still a mid-life addition, it was also a mid-life addition for Sony, Sega beat Sony to the punch, and Sega even added analog triggers. However, unlike Sony, Sega failed to realize that 3D gaming would be better served with two analog sticks. Worse, Sony’s twin sticks synergized well with having four shoulder buttons, while the decline of the fighting game market took away much of the benefit of having more than four face buttons. The Saturn pad and Sega’s legacy would be relegated to the past while the SNES-inspired Playstation Dual Analog design would be cemented in the continued evolution of game controllers.
After that, I don’t know what Sega was smoking, but it must have been whatever was left in the dumpster after the binge that led Nintendo to design the N64 controller. While the N64 controller looked like a drug-induced nightmare, there was a twisted logic to it and it worked surprisingly well for many N64 titles. The Dreamcast however was a flat out downgrade of the Saturn controller, “justified” by Sega instead going all-in on its ill-conceived VMU gimmick.