Carakav:
I need a flexible system for it that can cover everything from mundane skills and survival, to super-hero like powers/beings, and strange Call of Cthulu-style horror.
GURPS. It can handle every single genre you're wanting to mix.
Form your list and other mentioned:
M&M: I have no experience with this system.
Savage Worlds: Is fiddly. It can do those genres, but to me it feels like it's best not to mix those genres, otherwise you have PCs overshadowing other PCs very easily. Can be fine if you know your group and they can reign themselves in (ie Superman reigning himself in to let Lois Lane shine).
FATE: Theoretically it's also perfect, but where GURPS can do the
gritty survival very well, FATE will struggle to ever feel desperate. It's a narrative system so the fidelity can be pretty low at times. Works fine in PbP, beeter even for GMs who might not be 100% familiar with the rules as you have more time to think about/look things up.
Palladium/Rifts: Rifts is very, very, very fiddly and power-creepy. Each release expansion tried to up the power game over the last book. It has it's fanatics though, so it must be doing something very right (for them - I personally can't stand Palladium rules).
GURPS: It's perfect, it's the best, it's a bright shining star in a sea of dim bulbs... I'm of course only mildly biased. It can do gritty
very well, so well that most people don't think it can do high action or narrative play, but it can pretty easily... the GM just has to know how. And... well... there's the rub of GURPS in a nutshell, it's a toolbox to plug add-ons to the base skeletal system and flesh out your genre rules, but all that work is very much front-loaded on the GM. Now there are some supplements that can lighten the work load a little,
After The End is the line for post-apocalypse games,
Action is there with rules for high-octane action filled games,
Powers is there for super powers,
Fantasy and
Magic and
Thaumatology are there for branching in to the mythic... see what I mean? GURPS kinda needs a GM who knows it and knows when to turn rules off and on to fine tune the system to run the way they want it to.
But if you want, I've got some free time, I'd be happy to help out, wrangle rules and stuff for you, point out what rules sets you might want to use and what you'd probably want to turn off.