Yeah, my recollection is that full plate was hard to find even when non-magical. Plate mail was kind of common, though, just as you say.
What do you think about allowing the full DEX bonuses for medium and heavy armor, but at the price that one's weapon proficiency must be in heavy or versatile martial melee weapons (no finesse weapons, not even via feats)? Would that be enough of a restriction to make STR-based fighters a viable alternative to the DEX-based fighter, or no?
Well, ORIGINALLY, in D&D, and this extends into 1e, there is no 'plate armor'. AC2 is the best natural armor class possible in the game, and its platemail. There are PLENTY of magical platemails. DMG P123, table III.F ARMOR AND SHIELD. It shows chain mail as 11% of all results, +3 being max enchantment. Plate Mail is 19% and up to +5 (and includes the special type Plate Mail of Etherealness). There's also a 4% chance of a cursed Plate Mail of Vulnerability. Shields make up 27% of the distribution, with +5 being the best (and a 3% chance of a cursed shield -1, missile attractor). There are quite a few splint and scale armors and a few others, with light armors never going above +1.
Plate armor (or plate mail depending on which book you are reading, the terms seem to be interchangeable) costs 400gp, so you will not be buying it before starting out at level 1 (unless the whole party really really badly wanted to chip in, which makes no sense). Banded armor is the next best, and is only 90gp (Splint is equally good, but heavier and 10gp cheaper). Banded is only 1 AC short of plate, so that or splint is the 1e starting armor of choice. Later books added in 'Field Plate' as an armor type, at some high cost, like 2000gp or something, and 1 AC better than plate (AC1). Few characters ever bothered, as by the time you had 2000gp to waste on a point of AC you probably had magic armor anyway! Magic armor in 1e has NO ENCUMBRANCE! This makes it rather stupidly better than Field Plate, which is never going to come up magical at random (though there's no reason it cannot be). Field Plate is also custom made for your PC, nobody else can wear it, and you can't find it in a usable size (though it is ambiguous if the "one size fits all" feature of magic armors would apply here).
The upshot being, VERY few characters will be forced to start with under Splint armor. Fighters start with 5d4x10 GP, so averaging around 125gp. Banded armor, a longsword, and a small shield is 115gp, leaving 10 for a kit of basic 'stuff', which should be sufficient to get you started. If you roll crappy, you might need to limit yourself to 75gp chain mail, or even the crappy 40gp Scale armor (AC 5 with a shield). In other words, your cash could definitely cut into your AC if you had some bad luck, though one or two of the other PCs would probably be wise to front you 10 or 20gp to get you a better AC. OTOH some players would rather hire a couple spear chuckers instead, go with ring mail, and hide behind them, lol. In any case, getting to level 2 fighter is almost sure to involve collecting around 1500gp, so 400gp for plate mail at that point is pretty much a given. VERY few PCs that could wear it would not equip plate ASAP.