I started a similar post a couple years ago, so I know (and agree) with where the OP is coming from.
After several pages of that thread (which is why I haven't read past the first page of this one--I'm assuming it continued in a similar manner) and discussion with a friend in my group who took the opposing position, I finally came to a conclusion:
The disagreement is philosophical rather than mechanical. There are different positions being presented over what the purpose and goals of medium armor should be in the game. Debating the means to achieve goals is fruitless when the goals are different.
In order to understand my (and I believe the OP's) point, and provide useful feedback, you have to understand and respect the validity of our philosophical position on armor categories. I'll explain it as best as I can.
1) All armor categories should function in the same way. It should not be true that Light and Heavy armor fill one design role in the game (such as providing the best options for dedicated combatants), while Medium armor fills a different one (such as providing a good option for other types of characters). The remaining points will explain what I mean by the different categories functioning the same way.
2) The heavier the armor category, the better the armor should be. Heavy armor should provide better protection than Medium armor, which should provide better protection than Light armor. If a class gives you Medium armor proficiency, while another class only gives you Light armor proficiency, that should be saying that class has better AC (all other things being equal).(Because Heavy armor's protection caps at 18 an Light armor caps 17, there are levels where either Light and Medium, or Medium and Heavy, will have equivalent protection--but Light shouldn't be better than Medium, or Medium than Heavy, except in cases of Stealth-focus, see #3b.)
3) This should be true of all characters who are focused combatants. Focused combatants are assumed to raise their primary attack stat (Strength or Dexterity) to 20, and not to invest in the other. (It doesn't matter whether you actually do that with the character or not--I don't always myself--but the design needs to make that assumption).
3b) This means that you should be most effective wearing the best armor category you are proficient in. If you are proficient in both Light and Medium armor, you should be best off wearing Medium armor. If you are proficient in all three categories, you should be best off wearing Heavy. The exception in both cases is that if you are interested in using Stealth, you should sometimes use a lower category of armor (ie, Light if you are proficent in Medium or Heavy but have great Dex, or Medium if you are proficent in Heavy but have decent Dex).
4) This should hold true across all levels of play. And this should especially be true with maxed stats, since characters will likely get there before high level, and then be that way for the rest of the game. If there is any virtue to the idea that you can ignore levels of play above 15 because most campaigns stop before then, this doesn't work here because you max out much sooner. It should not be true that Medium armor is a good option until everyone has maxed out stats, at which point it becomes obsolete for focused combatants.
5) Medium armor should not require an extra resource investment to fulfill this function and purpose, since neither Light nor Heavy do.
Now, if you disagree with any of these points, you are coming from a different philosophy entirely, so there isn't any sort of useful debate about the means/mechanics to implement the goals. Now, there can be useful discussion about how to meet the goals of this design philosophy while also meeting the goals of a different one (such as that Medium armor's purpose is to provide good options for those who aren't dedicated combatants), provided those philosophies are not actually opposed. "It doesn't matter if..." is a statement that a factor isn't relevant to your philosophy, but not necessarily opposed, so a compatible solution for it along with other philosophies is possible. "I don't want it to be so that..." is a statement of opposition, so there isn't a possibility of a compatible solution.
After the discussion in the past, we derived a fairly simple solution that works for this philosophy. For those with other philosophies who might have people in your group who share this philosophy, you can decide if it fits both or not.
Solution: Medium armor adds either Dexterity (Max +2) bonus or Constitution bonus (Max +2) to your AC, whichever is higher.
This solution works because because adventurers in general, and focused combatants in particular, almost always have a 14 in Constitution, meaning they get the full benefit of Medium armor without any extra investment (#5). With that +2 bonus to your Medium armor AC for everyone (#3), the Heavy > Medium > Light scale (#2) holds across all levels of play (#4), fulfilling the overall goal (#1)
I also recommend changing the Medium Armor Master feat to provide a straight up +1 benefit to AC in Medium armor. That way your Medium armor can provide equal or (usually) better AC than Light armor for a Stealth-using character who is Strength-focused. This is particularly useful for some ranger archetypes, but it also works for any other Strength-focused character who wants to wear good armor without a Stealth penalty. This, however, is not a necessary change to make the solution work (allowing Con bonus to Medium is the only necessary change); it is simply a way to allow the Medium Armor Master feat to actually provide a useful benefit to Medium armor in the same way as Heavy Armor Master provides a benefit to Heavy armor, by providing an option for improvement that Light doesn't, providing more emphasis on Medium being better than Light.
Other Ramifications: Applying this change to monsters (and you should) means there are a few monsters whose CR's might jump up. To keep it simple though, I wouldn't worry about it unless their AC jumps by 4 or more, because that is the minimum amount necessary to guarantee that their CR will increase by 1. As far as I am aware, there are no monsters in the Monster Manual whose AC will increase by 4 or more with this change. With an increase of 2 or 3 there is a (theoretically) 50% chance that their CR won't increase, and with an increase of 1 there is a 75% chance their CR will not increase. I think ogres are the biggest change with 3, and duergar are the runner up with 2. So nobody necessarily even has a changed CR. For the vast majority of monsters, no change to CR is necessary--and for many monsters in Medium armor, no change to AC even takes place.
Visualization: So what can the Con mod bonus option represent in the fiction? First, let's think about what the Light and Heavy armor defaults can represent. An unarmored character basically is dodging blows with the Dexterity, and Light armor is effectively just a slight boost over that. Essentially, you can see it as stopping glancing blows from penetrating the armor. Dexterity mods (including penalties) don't apply to Heavy armor, so you can see its bulk, weight, and fit as having a stabilizing but limiting effect on your finer muscular reactions--the kinds where someone with high Dex might be able to dodge well, but someone with low Dex might be more likely to dodge the wrong direction. Heavy armor still allows the gross muscular reactions of some dodging (which is why certain normal tactical situations can provide advantage or disadvantage on attacks against you), but not enough for your Dexterity to matter. Medium armor can function the same as Light armor, but if your Dex is less than 14 but your Con is 14 or higher, you can see it as your armor blocking the worst of the hit, and you being able to take the remainder of the force well. The blunt trauma, or cuts and pricks, that make it through aren't enough to cause damage to you if the hit missed because of your Con mod. It's not perfect, but it's about as strong as any justification for Heavy armor not being penalized for low Dex.
