Also, if the design intent was to allow all characters wearing armor to have the same AC for the same investment, it seems like you could have done it much more simply based on just having, mechanically, one thing called 'armor' that you could reskin to suit your fancy. I guess maybe it would not have 'looked like D&D', but it would have been more honest.
While all avenues of AC can be close to comparable, it's not quite the same investment.
With Heavy Armor, all a character needs is some coin and they can have AC 18 at any level, 20 with a shield. Best if you also have Str 15+, but it's not required if you're fine with losing 10 speed (or you're a dwarf).
Medium Armor and Dex 14 the best you can do is AC 17 (19) without taking the Medium Armor Master feat, and then you need Dex 16 Dex to match the heavy armor wearer. This is essentially 2 ASIs worth of investment for 1 extra point of AC.
Light Armor is typically on primary Dex users. Assuming Dex 16 at 1st and the best light armor, that's only AC 15 (17). They need to reach level 8 in most cases and use all ASIs to increase Dex to reach parity with Medium Armor, and they'll always be 1 point behind Heavy.
For only 200 gp and no hard stat requirement (Str 15 or dwarf if you don't want a speed penalty), the heavy armor user gets AC 17; a medium armor user has to purchase a 750 gp suit of half plate and have a Dex of 14+ to match that, and a light armor user has to reach Dex 20.
Characters that have a feature granting 13 + Dex are generally better than Light Armor, with potential to match Heavy with a Dex of 20, but often characters with such a feature want their ASIs for things other than boosting Dex.
Barbarians and Monks can achieve a natural 20 AC but it requires having 2 stats at 20 each. Unless you rolled really well, that's probably going to take almost all of their ASIs.
TLDR is that Heavy Armor allows you to achieve close to the best AC in the game with little to no investment (assuming you gained the proficiency via class or race). It isn't until the very high levels where barbarians and monks finally start to surpass your AC, and most other characters will never even reach parity.
I'd say that's a pretty good reason to use Heavy Armor if you can.