A really good site to see how PBP is handled is rpol.net.

Greetings, Storytellers. I started playing TTRPGs in 1979 with D&D in a box. I still have the 20-sider from the set of cheap plastic dice from that box. In the early 80's I branched out and my first non-D&D system was Shadowrun. I played a Burned Out Mage with a pair of shotguns implanted in my arms. I moved on to everything Palladium, TMNT AND RIFTS being my favorites. I discovered the Hero System aat the very end of 3rd Edition, but didn't buy any NEW books until 5th Edition. And even though the system is math heavy during character creation and then regular play here and there, I found that it let's me do EVERYTHING I ever wanted to do in a TTRPG. If I can imagine the potential character, I can build it. Period. I often joke about creating a floating sentient rock, named Phil, until I made him, It? As long as you stay within point limitations and the GM's limitations, your character will ALWAYS be balanced against every other character made under the same constraints, (before adding XP). My approach to TTRPGs is cooperative storytelling. Of course there needs to be conflict here and there to add flavor and satiate the "Battle Gods". There are NO "random" encounters. Every encounter needs a purpose, even if that purpose isn't attached to the current adventure. If the players delve deep enough, they will discover that purpose and maybe spur on the next adventure.

I started playing D&D back in 1979, the good ole boxed edition where if you played an elf, that was your race AND your class. Elves where, for the most part, fighter/wizards. I've watched the game change over the years and get more and more expensive. I often joked that if you tried to read the 1st edition Dungeon Masters Guide, from cover to cover, you'd go insane. Yet I return to that old book time and time again for the charts and info packed into it. I stopped supporting D&D when they vomited out 4th Edition and haven't looked back. Everything since 4th has been an attempt to turn D&D into a video game played at the table. I have since discovered dozens of rules systems that are just as good, if not better than ANYTHING put out by TSR/WOTC/Hasbro. Palladium Fantasy, is as close to D&D as any system got, back then. In many ways it was better than even 3.5. My current favorite system is the Hero System. Character creation is loaded with LOTS of math. Nothing that a calculator, or the Hero Designer software can't handle. The Hero system is a set of rules that help you create ANY genre of role-play. There are some templates to apply, but nothing is set in stone, except by the GM. It's a point system where the GM gives you some parameters you must fir within. At that point, EVERY character is balanced against every other character from the bottom, up. If you want to play a floating sentient rock named Phil, you can build it with the Hero System. Your imagination, the rules, and the GM's build parameters are the ONLY things holding you back. I currently use the 5th Edition rules. I invested a LOT of money into those books, and I don't agree with most of the changes that they introduced in 6th Edition. I'm currently building an anthropomorphic cow, for a Champions (super hero) campaign. She has martial arts and elemental air powers, sort of Air Bender-ish. She's seven feet tall and has a human body. The only outward signs that she is not human is her cow-shaped head, and hooved feet. Psychologically, she is kinda like Superman in her "Boy Scout" attitudes, but is far more rooted in the realities of everyday life for the humans she tries to protect. In the world she operates in, Meta humans and creatures are common enough that the government created a two year training and licensing program that allows the license holder to operate as a Free Agent "Hero". It's like basic training with deep dives into Law and crime prevention. It helps an "Agent" hone their powers and learn when not to use them.