I'm intensely old school? That's news to me!
That has been my impression, yes. Since I don't know you personally, that impression could be entirely false.
There really isn't much design space with weapons. IIRC 4e had more features weapons could have, so there could be a greater number of different combinations. It was kinda nice, though I recall there was some balance issue with that too. It is kinda a matter of taste whether you want weapon selection to be a important minigame, or mainly just an aesthetic preference with some broad strokes mechanical categories. But I think rapier is OP. Not significantly, but I think the game would be better without it.
I can't recall any serious imbalance issues with 4e's weapon properties. The only real "issue" was the ways weapons interacted with feats, but that was intended as a space for players to explore if they were interested. (There were many ways to build a more powerful character, and "specialize in a specific weapon type" was only one path.) For example, long swords were classified as
heavy blades (basically, swords that use Str) while rapiers were classified as
light blades (finesse bladed weapons: daggers, sickles, short swords, etc.) The two categories were similar, but some magic items could only be one type, and some feats only applied to light blades, not heavy blades.
Now, I think that 4e might've had too many categories, and hung perhaps a bit too much off of them. But a streamlined, focused version could do a lot of good. For example, we could have:
Blunt (hammers, clubs, maces, staves, etc.)
Heavy blades (big swords, glaives, scimitars, scythes)
Light blades (daggers, rapiers, sickles, short swords, etc.)
Polearms (hopefully self-explanatory)
Axes (ditto)
Spears (spear, trident, javelin, lance)
Maybe Flail (flail, morningstar, spiked chain, nunchaku, kusarigama, etc.)
You could potentially drop Fail and fold Spear into Polearm if you prioritize parsimony over all else, but I think seven groups makes for a nice number--not too many as to be unwieldy, not too few as to be limiting. This is
essentially already present in the design of several feats, such as Pole Arm Master and Great Weapon Master, so it's not like this is doing much of anything except making certain feats and fighting styles simpler to use.
Between having feats to make these various weapon groups more interesting, adding in weapon properties as I mentioned in a previous thread, and reviewing the weapons so we can address issues like the
completely pointless trident or the "short sword and long sword are kinda meh compared to rapier," I think you could massage 5e into being a system where weapons have interesting depth IF you care to invest into it, but even Fighters can just get by with Fighting Styles and basic weapon choices.