I would avoid advantage/disadvantage because those play into other features. For example, if you ruled finesse weapons have advantage against no- or light-armor, rogues would always be able to use sneak attack. I think a flat 1 (maybe 2) point to attacks would be better.
You'd also want to avoid making rulings based on categories like "finesse", because some of those weapons are going to be great against unarmoured targets (Scimitars, for example), and others would gain no particular edge (Daggers). It's more like slashing is best vs. light/no armour, piercing is designed, generally, to work around armour, as is crushing, and "military" weapons are more designed to deal with armour than non-military (which is somewhat close to the simple/martial categories in D&D).
And really it begs for more nuance. Hitting unarmoured peasants? A slashing sword will gain more from that scenario that a Battleaxe, which also does slash. Dealing with a dude in full plate? A warhammer is likely going to have a significantly better time than a mace. Chainmail got you down? A normal spear with a relatively broad head isn't going to compete with, say, a war-pick (which will probably go through it like it isn't even there). But Battleaxes are martial, as is a Longsword.
And we haven't even got to monsters yet, which probably the majority of attack rolls will be against!
I think you'd really have to start over, and characterize all armour types, and all monsters as having a specific kind of armour, then give weapons additional characteristics so they can match up against the armour types.
Then if you were me you'd do another pass and weapons under a certain length would give you disadvantage when fighting monsters over a certain size - except if you were climbing on them...
And basically by the end I'd be writing Dragon's Dogma, the pen and paper RPG.