Games are built around a core mechanic. This core mechanic is what pushes forward the theme of the game. In the past, core resolution mechanic and core mechanics were different. Games used several systems together to give a feel for the themes they were built around. More modern games are built with some games - particularly indie games - having that theme in their core dice rolling mechanic.
When choosing a game, it is important to consider these mechanics. After all, running a horror game where there are supposed to be real consequences for fear and acting in fear is going to be a lot harder in a D&D game than it is in FATE. Why? Because D&D has no real built in mechanics for handling fear aside from the combat focused "Frightened" condition. However, FATE's Aspects and Consequences allow it to come up and be meaningful.
One of my favorite things to see in games - especially newer games - are design aspects that show the person knew what the game was supposed to be about, and then built the minimum number of systems around that so the game can be focused on it. Conversely, one of the things that always makes me sad with some of Fantasy Flight Games' RPGs is how many things they build on top of their core feel mechanic that just...kind of gets in the way. If nothing else, it makes the game a lot harder to teach.
Now, this doesn't mean you can't run things a system isn't built to handle with that system. FATE can do dungeon crawls. You can run a horror game in D&D. If your players are into it, it will work just fine. It just means that there is going to be more to do on your end and on the players end, because the system doesn't have it baked in.
This is also one of the problems I still see with 5e D&D. The game says it is built around 3 pillars: Combat, Exploration, and Social Interaction, and some of the classes have stuff to help with that in them but by and large the game is still just built around combat. There isn't anything there for social interaction aside from "Roll Persuasion/Deception/Intimidation" or "Roll Insight." There are no systems in it for exploration, for how the PCs scale a mountain or explore the deep seas. You can do those things. You can put them in your adventures. But the system isn't built to support it. Hell, XP is still only given for combat and encounters, with only a few suggestions for giving it out for mission objectives or other things.
Conversely, the other way to look at this is when making adventures for your game of choice, what does it lend itself towards? If you're running Edge of the Empire the dice system lends itself to all sorts of crazy adventure movie stuff with lucky and unlucky breaks able to happen. Masks: A New Generation can do everything from social engagement to combat using the same moves and consequences because all of it is built around the emotions in play for the character. 7th Sea 2nd edition is absolutely great for letting PCs feel heroic reinforced by the "Roll then choose" method of dice resolution that means the PCs are only failing by choice, and generally succeed at what they do - they just may not be able to do the thing they want to do in that moment.
Playing to these will help you, as the system will more naturally take the weight.
Run what you want. Run what you want in whatever system you want. But consider what the system can do for you, and what it can't do. If nothing else it'll help you know where you need to have more prep to carry the load.