The difference is that Dexterity already goes toward so many things, while melee is the only real thing that Strength does. If you remove Finesse weapons, then Dexterity is still a very useful stat. If you remove Strength-based melee weapons, then Strength becomes the universal dump stat.
At my tables as a player and as a GM, we track weight, encumbrance (variant), and carrying capacity. With that alone Strength is not a dump stat. Even as a scout, I had low strength and quickly realized I couldn't even loot the bad guys because I couldn't carry much more than my own pack and a few hundred coins. So if your not tracking coin weight, encumbrance
PHB p176 for that horrible 10 speed, limiting what the party can carry based on the lift/carry/drag stats in the PHB p176, or using grapple/shove PHB p195, using athletics (strength) for climbing, swimming, & jumping
PHB p175, or strength tests for trying to force a stuck doors and break locks or even strength as an alternate intimidation (strength) as suggested in the box "Variant: Skills with Different Abilities" PHB p175 … Then sure strength is only used for melee, but not by rules but because your choosing to ignore it and the rules to which it applies then wondering why no one is using it....
I'm in favor of having one stat to govern all melee combat, but I would want to call it "Strength", for the sake of tradition. I don't really see the benefit to allowing alternate descriptions, such as fighting purely with finesse such that Strength becomes a non-factor. To me, that's a very slippery slope, which inevitably leads back to 4E-style using Constitution to attack. You need to draw the line somewhere, and I see nothing wrong with limiting melee to Strength-only.
Sure, that's a valid opinion to take. However, I feel like stripping dexterity melee out is like saying a light weight boxer is not an athlete because heavy weight boxers exist. Their is something to be said for skill and speed vs power as separate fighting styles. I don't want ninja based on sumo strength. Thematically I like the separation. As I said I don't want one stat, because as you said its a slippery slope. I like skill (dexterity), strength, and toughness (constitution) as my variable on any fighter. Less seems like a lose of variety and more seems nit picky.
That is absolutely opinion and preference. If I have a complaint its that a longsword is strength weapon in D&D even though its a back balanced cutting weapon make for speed as skill ….so I just have to remind my self its a fantasy game somethings are not perfect. You have your desire and I have mine. So what's wrong with players liking the skilled fighter stereotype over the strong brute stereotype? The mountain vs that fancy guy with a stick?
I know it's just a matter of preferences and expectations, but to me, a monk is a Strength-based fighter in not-necessarily armor. The bar-room brawler isn't even a concept worth entertaining, because it would be so pitifully ineffective when compared to a fighter in armor, and monks are amazing because they actually can perform comparably with their bare hands. Simply using the same stat to hit with is not sufficient to conflate the two, anymore than a paladin is like a warlock because they both use Charisma.
I agree with your first statement, but to me, a Monk is like a Ninja skill (aka dex) and no armor and Strength with no armor is Conan the Barbarian and strength with armor is The Mountain.
When you're talking about light armor, in D&D, that's the stuff that rogues and ninjas wear. It's "studded leather", or some other nonsense that gives minimal protection, but is sufficient to fulfill the mechanical role required of it.
Sure they could of put the gambeson as a better descriptive match and look but if I am thinking monk I am thinking like Bruce Lee without a shirt and the wisdom dexterity combo of reading your opponent and dodging then punching in the week spots is the monk I think of. Like Ip Man vs Mike Tyson the strength based barbarian.
https://www.youtube.com/watch?v=6HvnRvUSHr4
They are not the same and I like the separation in D&D. I have no problem with all my players wanting to be Ip Man/Jet Lee ninja rogue monks if they are not interested in being Conan the Barbarian or the Mountain. To me, it is a narrative choice and the since we do use all the strength rules I mentioned above and we do some small level of party planning, we always have a couple of players using strength shoving people around, carrying gear, and grappling prone enemies so the party can all gain advantage on attacks until the enemy breaks free. We also usually have a healer and tank at the start of the campaign. Though I healer left for IRL reasons and we didn't change the current party because of that we just got more careful. That said we have a paladin tank, so he can do some amount of healing anyway.