I think the problem with it is how mysterious and arbitrary the rules of a fighting game can be. When I used to practice guitar and I couldn't play a particular line, it was easy to place the blame on myself. Playing a guitar (at least, playing it poorly) isn't rocket science: you fret the note, you pluck the string when the beat comes, repeat. if you fail at steps 1 or 2, it's your fault 99.9% of the time. Fighting games are different. I remember showing the trial mode from Super Street Fighter 4 to my girlfriend. She was interested in learning how to play and was going along fine until she reached a trial where she had to link(word play!) two normal attacks. I forget which, I think a crouching medium punch to a standing medium punch with Rose. Something that seems simple but actually requires precise timing. But the problem isn't her timing (my girlfriend is onethousand times the musician I am), the problem is the game doesn't communicate what the timing is at all.
I was thinking about this recently when i was linked to a video of world famous violinist Itzhak Perlman talking about Practicing. While he's talking about playing music, I think he knows at least a little bit about picking up a skill. And I think that's probably the problem with game designers for fighting games. I doubt many of them have taken a fighting game seriously and don't know the trials and tribulations in becoming competitive at them. For that matter, I don't think most people who are actually top tier competitors at fighting games know what it takes to become good. Most of them just went to tournaments for all of their lives until they became amazing and then talked shit about their talent.
I think all of this should change, personally. And I think the biggest obstacle is the trash they call training mode in modern fighting games. Because of the way training mode is designed, if you aren't already dedicated to becoming the best, have access to frame data, and a lot of time to really explore the system, you'll spend a few minutes in training mode when you get the game and then only come back maybe to check how two moves interact with each other.
It doesn't have to be this way! I think to date, the best training mode options I've seen have been in Guilty Gear XX and Street Fighter 2 HD Remix (Fun fact: the creators of each of these games were very into fighting games at different points in their lives). Guilty Gear gives the players a treasure trove of options to break the game's system and push it to its limit. SF2HDR gives players an option to view hitboxes which is unheard of (for some reason). But these are still not quite there. What would I add? I can think of three options off the top of my head to improve the efficacy of fighting game training modes.
1) Slow down
This was actually featured in Super Smash Bros' training mode, but no one picked up on it (maybe cause it's technically not a "fighting game" but a "competitive action game" (whatever Nintendo)). I'm not sure why. This obviously plays into Itzhak's idea of practicing slowly (hope you watched the video). Some fighting games require you to do a move and then follow with another move, sometimes within 1/60th of a second. That sort of timing is certainly possible (I've watched plenty in real life matches), but without the ability to slow things down and really figure out the timing, only the most diehard fans can be competitive.
2) Color Coded hit boxes to show when cancels/links are possible
I think this one is pretty obvious, but would actually be really hard to implement. Mainly because if a character had one cancelable-move, it would be difficult to tell which other moves would be cancelable. But assuming a solid solution could be designed (again, this is off the top of my head), this and the previous feature would make combo discovery and composition a much easier and enjoyable experience.
3) Programmable two way (or three way!) mixups
This one would appeal to the more technically inclined players I've talked about, but I think it would be great for anyone who wanted to get serious about fighting games. For the uninitiated, when a player has the advantage they often put the other player in a situation where they must make a guess. If they guess right, they usually get out free, if they guess wrong, they receive damage and usually are put back into the same situation. These situations are hard to practice on your own, because the CPU can only record one set of inputs. If a training mode allowed two recorded inputs that would randomly be switched between, a player could train their reactions against the world of mixups, instead of just whatever they run into from their friends.
Look at all that, right off the top of my head. If (when) I end up making a fighting game, I'm going to make the greatest training mode in the world.