All depends what you think the problem is I suppose
I believe that BookBarbarian was referring to the Versatile property of weapons not being used because they are already strong as single-handed but weak as two-handed because they lose the use of a shield, but don't compare well to dedicated two-handed weapons.
The longsword itself is excellent. Longsword = bastard sword.
• longsword, 1d8 slashing (versatile 1d10) + Strength
There is a need for a socalled knightly arming sword.
This is an ‘agile’, ‘cut and thrust’, sword.
It is the archetypal sword.
Call it ‘sword’.
• sword, 1d8 slashing or piercing + Strength (finesse Dexterity)
Note, the sword is statistically identical to the rapier, except the sword can alternate slash and pierce.
5e definition of a longsword already covers the arming sword, falchion, dao, many sabres, bastard sword, macahuitl, 'Viking sword' etc.
Some examples are optimised towards two-handed use, and some are built for mostly one-handed use, but 5e isn't really granular enough to pick out that level of distinction.
The 'arming sword' isn't heavy, but it still has appreciable weight, and requires strength of both shoulder and forearm to use with speed and control. It was also used two-handed fairly regularly in halfsword techniques when a bit of extra oomph was required and a shield wasn't used. - To me, that puts it squarely in '5e Longsword' territory.
While we are at it. Also add in katana.
• katana, 1d6 slashing (versatile 1d8) + Strength (finesse Dexterity)
Note, the katana is the only weapon so far that is both versatile and finesse.
Here, the knightly ‘sword’ is slightly better than the katana.
I feel it is ok to give the sword plot protection.
(Like giving lightsaber plot protection in a Star Wars game.)
But if someone wants to make the sword and katana equally good,
maybe give the katana a minor benefit for being razor sharp.
The 5e definitions of Finesse and Versatile don't really make sense together. The point about putting two hands on a weapon is to improve control and power though increased leverage. Finesse weapons do not require leverage since that is a function of strength.
If you want to have a finesse version of the katana, just give it the stats of the scimitar - which it already quite closely resembles physically.
Elf. Three solutions.
1. A Dexterity elf gets an ‘elegant’ swordfighting style that applies finesse to any longsword.
2. The finesse ‘sword’ now exists. This is actually the sword that the elf culture was celebrating.
3. Eliminate the race proficiency with longsword or any sword. Who cares really?
4. Leave as is. Elves get standard Longsword proficiency.
(I tend to swap the weapon proficiencies around anyway, since in Eberron, one of the major elven nations tend to emphasise scimitars and shortbows, and the other elven subrace; the drow go for blowguns, boomerangs, and long knives rather than the traditional elven weapons.)
A punch is about as ‘finesse’ as one can get − and it requires Strength.
I'd suggest almost the exact opposite: a punch is about the least Finesse (in 5e D&D terms at least) weapon around. Its effect is almost entirely dependent upon the pure force it delivers, even when it strikes the few well-guarded areas on the body that are more vulnerable.