It's no secret that my two go to games are Savage Worlds and D&D 5th Edition. It's also not uncommon for more and more table top games to have some version of a meta mechanic, hell Fate is actually one giant meta game. However, for the sake of brevity and the fact that these are the systems I'm know the best these are what I will cover. For the odd chance that any of you aren't aware of what either or even both of these mechanics are I'll start with a quick run down.
The two biggest additions to 5th Edition over earlier editions are Inspiration and Advantage. Advantage is something that HAS existed earlier, but now rather than a bonus it's been broken down into rolling twice and taking the highest (or lowest in disadvantage). I brought up this other mechanics because all Inspiration does mechanically is grant Advantage. The really trick in the mechanic though is how you gain Inspiration. It's basically handed out at the whim of the DM (or GM) for anything from attempting something awesome, to making people laugh, to (how it's most meant to be awarded) playing into the flaws and quirks of your character. Sure since it's awarded for role playing you might say this isn't meta, but the thing is you can then hold onto that inspiration to use when you need it. This includes a moment that might have nothing to do with what gave you the inspiration in the first place. This means its a currency that exists outside of your character and does not need a line from cause to effect making meta.
Similar to Inspiration, Savage Worlds has a mechanic called Bennies. You get Bennies for basically the same reasons you would get Inspiration, but players also start with three of them off the bat and (unless the GM has put one in effect) there is no limit to how many a player could stock up on. While it doesn't give "Advantage" the core use of them is to let a player re-roll a check which as we said is basically the same as advantage just which choosing to use it after you know the first roll sucked. The other use is the "soak" damage letting a player spend a Benny and attempt to make what could be a killing blow or at least a serious injury be not as deadly.
With that little lesson in gaming is over now I can get into the nitty gritty of what I wanted to talk about, the first of which is weather meta mechanics deserve a place at your table. Personally I'm fine with them, however I do understand any argument against it. A meta mechanic by its very nature removes at least a bit of the player from immersing themselves in their character. It puts them in 3rd person mode seeing the game more as an author to a story than just living out that story.
In my opinion battle mats do the same thing. Let players get a aerial view of the battle field and in turn letting them see much more of what's going on than their character might. In that case a player can decide to ignore that information (though try and you might it's still there in your brain, you're just choosing weather or not to use it), but with Bennies and Inspiration the only way to ignore it is to just not use them. D&D Has at least a little bit of a "fix" for this as your character can flash back to the moment that gave it to them as a form of inner strength.
While this works from a narrative sense and is even a bit of a trope in writing I would like to ask, "If you didn't have Inspiration to spend wound your character still have looked back at that moment?" In stories this happens as a way to show how foreshadowing has come to fruition and in a way a GM could be very careful in when they had out inspiration so that the moments looked back on are always story relevant, but that's a product of good GMing and not the mechanic itself.
Bennies are actually even worse about this as the system describes them basically as luck or fate granted to the characters for being stars of the story and starting with three means they have nothing to connect to. A fun house rule I might want to try is to combine the interlude rules (A system for players telling backstories in character) with Bennies so that when they spend on they then must have a flash back to give them this luck. I'll be honest that idea JUST popped in my head while typing this, but I do think it could make for a much more narratively interesting story and add depth to characters. The issue being you can easily spend five or six Bennies per session so I would probably say you only have to narrate each roll so if you spend more than one Benny on a role you are only narrating once.
As for Inspiration I'm not gonna lie I'm toying with an idea that will let it get a bit more usefulness out of it similar to Bennies. One of the issues is if you already had advantage then you can't get further advantage. I've toyed with allowing each spend to just be an extra D20, but I'm not quite there yet. The closest I'm thinking so far is if you naturally have advantage and disadvantage (normally these cancel out now matter how many you have on either side) you can spend your Inspiration to actually have advantage. The other use is similar to the "soak" ability from Savage Worlds. In this case you can spend your Inspiration to make a Constitution check and whatever you roll you reduce the damage by. Yes I know this is powerful, but the make this could make (Without magic or being a 20th Level Barbarian) is 25 damage and while this seems like a lot at low levels it is next to nothing at high levels. This could also be a saving grace from the instant death that can so easily happen at the earliest levels.
Now with all these extra uses I also feel like inspiration will come up a lot more so I think I might tweak the limits on how much a player can have. It won't be the unlimited of Savage Worlds, but instead I'm thinking a cap of three and every character starts with one.
I was tempted to also talk about meta mechanics that let players change the story, but D&D uses a different resource for it and Savage Worlds (while using Bennies) has it as an optional rule so I figured that might be best left for another day.