Your analysis has a few flaws
Comparing Plate to Splint +1 is as nonsensical as comparing Studded Leather armor to Leather +1. Should Studded Leather be as rare and hard to get at Leather +1? What about Padded +2?
The reality is, in just about every DnD edition, magical armor makes little sense except for the best versions. (Chainmail +1.... why?)
Yes, the armor rules do not always make sense. However, the parts where they make the most sense tend to be at the points that are really relevant - the points that PCs will gravitate towards - the points where the best AC can be achieved for a particular build.
Leather +1 and studded share a lot in common, but they fail to have some key things in common that platemail and splint +1 do: They're at the top end of the armor spectrum and they've expensive. However, as you note, leather +1 is - in almost all circumstances - effectively equivalent to studded leather (although I would tend to argue studded is actually better than leather +1 more often than the reverse). It is a pretty bad treasure haul.
Regardless, the similarities between leather +1 and studded leather are not relevant here because they did not take steps to make both studded leather and leather +1 hard to acquire and they do not exist near the top of the armor range.
And at least Chainmail +1 makes some sense for a 13 or 14 strength character....
Before we start to throw in the 'but a high dex character gets almost as much out of studded leather as platemail offers' argument, a high dex character is benefitting from more resource allocation than just a suit of armor to get that AC. They generally have a high dexterity at the opportunity cost of having a high 'other' ability score. They're paying for it, just in a way other than just armor.
Second, even only counting Hoard treasure, you should have enough for Plate before 6th level. I had it at 4th level because the party chipped in. (Since not much else to spend it on.)
If you follow the guidelines for treasure hoards you should collect about 375 gp worth of money, gems and art objects per hoard. With 7 hoards, you get ~ 2600 GP TOTAL for the party by the you hit 5th. Some of that should go to cost of living, upgrading other initial equipment, etc... If an entire party devotes the vast majority of their resources to getting one member plate mail - yes, in a typical party you should be able to get there around 5th level... but that is not expected behavior in the game. Following the expected guidelines and a typical division of resources amongst a party, the typical/expected zone where platemail should be acquired is more likely to be where I stated.