It depends upon your choice of fighting game and the type of lag.
The history of fighting games is an interesting one full of accidents and bugs becoming features. Either way the amorphous mass that Street Fighter 2 eventually became maybe did not start what was to be the dominant trend but certainly cemented it.
Said trend is of large move lists of fairly complex actions, and very very specific timings for things all operating on fairly complicated hitbox models*. It is why some liken fighting games not to a contest of reaction but something like chess instead in that you have to out think your opponent. Fair enough, also one of the main reasons I find those games and top level competition quite boring (I imagine it would be for smash brothers as well but somehow the fools running every single one of the contests for it do not seem to have read the first page of a competition design manual which tanks it before we even get to that point).
*the absolute best form of armour is to not be there when the attack is made. As I type this were I a few hundred metres away I would have been splatted by a truck that went by, I only notice it here from the shadows it casts and its sound. As "that clearly did/did not hit" is a fairly easy way to break a game the devs play to it.
You may have seen smash brothers people discussing lag. It is actually a different type though.
There are two types of lag.
1) Conventional latency
2) Variable latency
1) Conventional is a result of a signal having to be encoded, sent, travel the distance, be decoded and then be acted upon. Still very much applies in the case of wires but for controllers of normal length in the hands of humans it is nothing.
This can be learned to be dealt with. You just pick different visual cues or develop heuristics. This has been known since the days of archery -- you account for the wind on the way and where your arrow tip is may well not be where it lands as gravity is a thing so you aim up a bit. Gamers by and large have been trained to do this for decades now -- you take a visual, auditory or some such cue and know to do some action at that time and it goes unsaid most of the time, or just called basic pattern recognition. This can land you in trouble if you go somewhere else with a nominally better setup renders your typical cues and reactions pointless. I get something like this when I play on a PS4 with a laggy as you like plasma screen and then go back to my blisteringly fast CRTs or TN panels.
Depending upon the reaction timeframes you have it can also cause problems. Example.
I know a fighting game such that I can tell you every move for every character, can do every move within human reaction times perfectly and I am on form that day. However in this game someone does a combo. I know it will result in one of two things happening but will require a different reaction to save me, I can do it under ideal conditions though. If my controller adds 10ms of lag to what is normally a perfectly workable 20ms window then I am down to 10ms which I might not be able to handle. For a fighting game that might be bad game design, for a multiplayer shooter then that's just life really and my sniper skills should be better or I should wait for my target to not be accelerating.
2) Variable is another.
Two sub types, possibly three if you want to get technical for some games.
i)dropped packets. If the packet does not get sent and the previous one sticks then you have a lag you can not reasonably account for when playing.
ii) a variation on the idea of dead zone and kind of what smash brothers peeps are concerned with.
Analogue controllers then. You might know they usually go between say 0 (top dead centre) and 256 (fully one direction), however if you have ever touched analogue electronics you will know it is very far from stable. To that end most controllers have a "dead zone" where it is assumed nothing is happening despite what the electronics say. If this dead zone, or the electronics covering it all, is a bit floaty and means something may happen a few frames either side of a better example of the same controller* then your frame precise timings (which some advances smash brothers techniques appear to require) may be thrown off.
*Gamecube controllers apparently being prone to this, hence smash brothers peeps seeking certain controllers, testing large numbers of them to find good ones or having a debate on whether a device to effect a better controller here, though still within parameters seen in the wild, counts as cheating.
iii) would be more of an argument when it comes to the 60fps discussion. Some say it gives you more time to react over 30fps, I say controllers can poll at whatever speed they like really and even do decoupled from frames these days. If you are coupling your input polling to your frame update rather than vblank or something on an older console/older game programming philosophy then you deserve your problems.
Also note that when talking about screens the decode latency and whatever the kids are using instead of a grey to grey speed are two different things and often confused -- you can have the fastest g2g speed but if it takes 100ms to get the signal from going in to going on the screen because it is processing the signal (why you may have game/PC mode on a TV and it may look worse than, or at least different to, normal modes) you are going to notice it. As CRT screens are very basic and their controls are mostly just analogue electronics (more voltage to the coil changing the beam shape and thus resulting screen layout) the former type was basically a non issue for decades.
Back to wireless controllers. Earlier radio stuff (infrared was popular for a while but owing to the massive downsides is no longer popular) was really quite bad, or at least massively variable with cheapo gamer stuff**. Massive investment and advances have been made in it since then and you can buy stuff off the shelf today for something around what you might have spent on lunch to play with as a kit for your arduino or something that would have made a greybeard proud 10-15 years ago. Still possible to hose it up mind you and I have not really tested all the weird and wonderful thing as Microsoft and Sony have millions to sink in testing this stuff and a vested interest to get right, not to mention with economies of scale they can sell the results to me for money I can more or less afford to lose.
**always with the asides today. I have seen the stuff that goes into making proper robust control electronics -- your $100 gamer keyboard fails, boo hoo. Your $100000 a day operation grinds to a halt because someone cheaped out on a control switch, that's a problem. None of the people designing for the latter camp gave the slightest damn about "cherry MX switches". It is all marketing.
All this said batteries dying is easily the most annoying thing for me when it comes to wireless controllers. To that end when I can plug a wire into my head/eyeballs I am so there.