The only official element that would limit acquisition of plate armor would be its exorbitant cost. Xanathar’s Guide to Everything (page 135) suggests that a party should have seven rolls on the Challenge 0-4 treasure hoard table and 18 rolls on the 5-10 table as it levels up. With average treasure rolls, with a group of 4 to 5 player characters, plate armor should probably become affordable somewhere around level 7, though that will vary somewhat depending on how much money you’re making outside of hoard rolls and how many people are splitting the proceeds. (This matches my personal experience, where the paladin in our party of 5 got his first set of plate just about the time we leveled up to 7.)
But I think you have it a bit backwards; even if you strike plate from the armor list completely, it doesn’t really change who wants to use their Heavy proficiency, and it doesn’t make heavy armor weak or undesirable. Medium armor is only particularly desirable if you have a moderately good Dexterity; if you don’t, you go Heavy if you can.
In general, you want to wear the heaviest armor you have proficiency for, unless you have a very specific reason not to — which usually means having unusual stats for your class (such as a finesse Fighter) or having unusual needs not normally associated with your class (such as stealth).
Anyone can find an armor to get to AC 16 pretty easily no matter what their stats look like, unless they’ve managed to submarine both their Strength and Dexterity.
AC 17, the maximum for all non-Plate armors, is pretty easily achievable if you care to go there, but it’s a lot cheaper and easier for Heavy armor wearers, provided they can handle the weight (which generally means a good strength score or being a dwarf).
In other words, if you hit AC 17, you’re doing well and have effectively maxed out your defense from armor. Plate is a little extra thing to stand above as the Ultimate Armor, and with a price-tag to match. But if you exclude it from your game or make it prohibitively rare, the fighter and paladin (and some clerics) are STILL going to probably wear Heavy armor, just because it’s the easiest path to AC 17 and matches their usual stats well.
It might be different if stat bonuses were easy to get, but since they aren’t, you usually need to focus stat increases on your core abilities, which doesn’t often leave lots of space to buff your Dexterity out of the Heavy armor range; and having heavy armor available is what allows these characters to distribute their ability points elsewhere.