The main problem with stamina mode that a lot of people don't consider are that most people make suggestions for all of the characters to have the same amount of health.
This simply doesn't work out too well, because it benefits smaller, more agile characters exponentially more than heavyweight characters. Because they have the same amount of health now, there is literally no point to picking a heavyweight character now. Not only do you lose their bulk in a stamina mode match, but their additional knockback is now pointless and does more harm than good. Because it makes the pace of the match reset back to neutral, while lightweight characters can probably Utilt chain them for about 5-10 seconds or more before a heavyweight can potentially break free, if at all.
Utilt chains and infinite combos are another worry for stamina mode. Things that don't normally exist in the main game because knockback scales with damage. This isn't the case in stamina mode, and therefore the chance to encounter problematic moves and/or tatics such as these increases. Take Fox's Utilt for example.
One way to help "fix" stamina mode for competitive purposes is to have different health values for the characters. Diddy, Pikachu, Sheik? They would have low health, due to the wide array of options and mobility tools that they possess. Bowser, King DeDeDe, Ganondorf? These characters should have higher health because of their limited mobility and relatively crappy moves within a stamina mode environment.
However, this is only a small fix, as there are a lot of problems with stamina mode still. Simply put, the game's engine was specifically designed to have knockback scaling in mind. And when you try to deviate away from the central design the game's engine was built around, you start to encounter more and more problems. What I mentioned only scratches the surface.
tl;dr the only way to potentially make this work is to give every single character their own individual health values based upon the amount of options they have available to them in a stamina mode format, and even then, this won't fix everything simply because Smash Bros was not designed to use traditional fighting game HP values over the damage/knockback formula the game's engine was built around.