Brief overview:
This particular hybrid genre is quite rare to find and very enjoyable for me. I think it can largely be credited to Ratchet & Clank for trendsetting and inspiring all the rest (even Jak 2 had used some of Ratchet's code after Jak and Daxter [1]: The Precursor Legacy didnt have any weapons). There are some older games that might have inspired Ratchet & Clank, such as Blasto, which will also be listed here. But mostly I'm focusing on games that replicate a similar style as the Ratchet and Jak series, sometimes on purpose. Please let me know if you have any additions to the list.
The List (main):
▪︎ Ratchet & Clank (series) ▪︎ Jak and Daxter (series) ▪︎ Protodroid DeLTA ▪︎ Akimbot and Pumpkin Jack (same dev as each other) ▪︎ Tamarin ▪︎ Automaton Lung (& sequel Automaton Heart demo) ▪︎ GearGrit (and the devs' other game Fadeout kind of) ▪︎ Cloudbuilt ▪︎ Onirism ▪︎ Metal Arms : Glitch in the System ▪︎ Ruff Trigger : The Vanocore Conspiracy ▪︎ Donald Duck PK (AKA PK : Out of the Shadows) ▪︎ SpongeBob SquarePants: Plankton's Robotic Revenge (yes, this is really a 3rd person shooter) ▪︎ Oddworld: Stranger’s Wrath ▪︎ Shadow the Hedgehog ▪︎ Spy vs Spy ▪︎ Blasto
I've heard recommended but can't verify if they fit the bill or not:
▪︎ Jet Force Gemini ▪︎ Stitch: Experiment 626 ▪︎ Smarties: Meltdown ▪︎ Insecticide ▪︎ Dr. Muto ▪︎ Destroy All Humans! (series) ▪︎ Robots ▪︎ One ▪︎ MDK (series) ▪︎ Earthworm Jim 3D ▪︎ Buzz Lightyear of Star Command ▪︎ Blinx : The Time Sweeper ▪︎ Cryght ▪︎ Mega Man Legends ▪︎ Fur Fighters
Similar gameplay elements either continuously or occasionally:
▪︎ Sly Cooper (series) ▪︎ Ty the Tasmanian Tiger (series) ▪︎ Rayman (the 3D games) ▪︎ Skylar & Plux ▪︎ Spyro the Dragon (series) ▪︎ Super Mario (series) ▪︎ Crash Bandicoot (series) ▪︎ Astro Bot (series) ▪︎ *Psychonauts (series) ▪︎ Futurama ▪︎ Family Guy Video Game! ▪︎ Jumping Flash ▪︎ The Gunk ▪︎ Splatoon (series) ▪︎ Sunset Overdrive ▪︎ Returnal ▪︎ ReCore ▪︎ Lost Planet (series)

Glad to see Onirism in the list. I still don't know where the title is from, but it's definitely an oddball and was an unexpected find with a ton of content.
Not exactly fitting the criteria, but in terms of using shooting mechanics and leaping between obstacles, an old indie I enjoyed is A Story About My Uncle. It even had surprisingly good facial animation for its time, especially since story isn't such a core element.
Thanks for the recommendation! Onirism is indeed a hidden gem, totally underrated. I believe the name Onirism means "dreaming/experiencing hallucinations while awake", related to the word "oneiric" (dreamlike) or the Greek "oneiros" (dream), and has been applied to psychiatry and Romanian literature. Makes sense since the game is so dreamy and surreal. :)