One of the best pieces of advice to give someone who wants to improve their writing skills is to read widely. The idea is that you will be exposed to more and more writing in a wider variety of forms, including those outside of whatever genres you might choose to focus on, all of which is useful to you as a writer because it will give you more tools to use in your own creative work. After all, the various writing tricks authors use, their various stylistic quirks and so on, aren’t limited to a genre. If you see something cool and interesting in a science fiction story, you can figure out how to incorporate it into a fantasy story. Or if you find a particularly interesting way of phrasing an idea in a piece of nonfiction, you can find ways to do similar things in your own fictional works. The more you’re exposed to, the more you’ve learned and can incorporate consciously and unconsciously. Which is also true of running tabletop games (and storytelling as a whole, but you can pretty much extend any of this advice into any type of storytelling with enough abstract thinking, so I’m going to stay focused). The more games you play or run, the better you are. This is fairly self-evident to most people since that tends to fall under the “experience makes you better at things” bit of wisdom. I’d suggest taking it a step further, though, and suggest that you play a wide variety of games rather than just sticking to the ones your prefer.
It will not take you long to find someone who is speaking disparagingly of other people in the tabletop gaming community. There’s people who get upset that some people only play Dungeons & Dragons, people who are upset that other people hack games to fit niches that already have games in them that they could be playing instead, and even people who seem like they just want a reason to be angry regardless of whether it is a good reason or not. I personally think people should be allowed to play whatever they want, be it the same single game for their entire life or never the same game twice. Life’s too short to gatekeep the Tabletop Roleplaying Game community based on what games people play. Still, even with that said, I think you’re really limiting your abilities as a collaborative storyteller if you don’t at least try other games. You don’t need to stick with them, of course, but playing them until you feel the system start to run smoothly will give you the same benefits as reading widely but for playing or running tabletop games instead of writing.
Probably the best example I’ve got of this in my own life is the way that The Ground Itself (by Everest Pipkin) has helped me do collaborative worldbuilding and handle shifts in time in a single place in every other game I play. The language and systems of The Ground Itself aren’t just restricted to that game because what they do really well is ask questions in a way that fosters creativity. Between the prompts, the sections handling changes in time, and the way it encourages players to build on each other’s work rather than around it, The Ground Itself’s language and phrasing can be an excellent little tool you could bring into any game you play when you wind up skipping around in time. The questions also do a great job of concisely focusing the moment on what changes and how it changes without letting it get too caught up in who makes those changes or why those changes happened. Those details are more interesting to explore through the more focused lens of most campaign-supporting TTRPGs, whereas coming up with former details, the what and how, can put players on the spot if they are trying to justify why their characters would do something or who outside of their characters would act on the world this way. Using this framing helps me stay focused on painting the broad strokes and leaving plenty of room for backfill later, as people come up with ideas for the game we’re playing evolves.
Another great example is everything I’ve learned about partial successes and failing forward from non-d20 systems. The idea of hitting specific goals and simply failing if you don’t reach the required number is a pretty core aspect of a lot of d20 systems (as are the ever-increasing targets required for a “moderate” success so that there’s always “risk” any time a player is called on to roll the dice) and a lot of tips on running those games include suggestions for giving additional information for exceeding the given target by a large enough amount. Few, though, suggest giving partial information on failures or making sure that there is still a way for the player characters to move forward even if they completely and utterly fail. Sure, sometimes the answer to a failed perception check is that someone just doesn’t notice something, but you can usually still find a way to give them enough useful information that they can make a decision about what to do even if the result didn’t give them all the information you had ready for a good roll. It requires a bit of reframing work on the part of the players as well since a lot of the d20 systems come with the assumption that failure means you get nothing, so they might get upset if you instead give them partial or wrong information and they eventually realize that you’ve done that when they’d assumed they’d gotten a full success.
The important thing to remember in all of this is that you’re not inserting entire mechanics from one game into another. You’re taking the systems you’ve experienced and the lessons you’ve learn, finding the interesting ideas or tidbits, and then bringing them into the appropriate places in other games. That could look like entirely inserting mechanics into a game, or it could just look like the GM reframing the way they’d ask a question or a player performing an action that prompts another player rather than moves their own character’s story ahead. The possibilities are endless as long as you keep an open mind and do the work to ensure you’re actually learning something rather than just regurgitating it. Your games will noticeably improve.