If you think about it, the importance of each side of the gamepad has also shifted to place much greater importance on the right side.
In the 6th generation(PS2, OG Xbox, Gamecube) with all 3 consoles using a dual analogue layout and the 4 face buttons made standard in the previous generation the importance of the D-pad was reduced dramatically. The rising popularity of FPS games on consoles also placed a much greater emphasis on right-stick precision than left stick precision, as movement never needs to be as precise as aiming.
Fighting games are one of the few genres where both speed and precision are required from your left thumb. 3rd person action games that don't involve aiming are also more focused on precise button combinations, something done by your right hand since it's on the face buttons. Outside of fighting games and some 2D platformers the D-pad is mostly used for secondary functions like weapon/item selection.
Even if you go back to the NES the D-pad being on the left made some sense, especially when you consider that its most popular and most famous game, Super Mario Bros, has a dedicated jump button. A trend followed by almost every other platformer since. You're performing the most important action with a face button press instead of doing it by pressing up on the D-pad like you would in a fighting game. This greatly diminishes the role of the D-pad and the dexterity required. Outside of some extreme speedrunning strats that require a perfect 1-frame release and 1-frame re-press you just don't need to do much with it. Just hold right, sometimes hold left for brief periods then hold right again. Meanwhile the action the game is built around, jumping, is handled by your right thumb.
Of course, the biggest exception here are racing games and flight games, both of which feature analogue movement as their most important mechanic and in most of them the gamepad control scheme still places that movement onto the left stick. So steering in a racing game with triggers acting as throttle and brakes and pitch+roll(or yaw) in flight games.