Bear in mind that in 5e, Strength does not require looking like a bodybuilder: its athleticism and ability to apply force as much as how much you can lift. Think Bruce Lee as much as Arnold.
I guarantee that, using a rapier, no amount of strength will make you more likely to pierce the armor so they feel it. some dexterous movements though, allowing you to slide into an opening in the armor, will work quite well. After that, it takes very little pressure for a sharp object to pierce human flesh.
It actually requires more strength to use a rapier effectively than a longsword. A rapier (as opposed to the whippy fencing swords used by Flynn & co.) is of similar mass, and is controlled to a large degree through the muscles of the hand and arm rather than having the leverage of both hands on the long longsword hilt. Many styles of fencing also emphasise athleticism - which in 5e is covered by Strength.
I am simply not convinced that being stronger gives you an advantage to "hit" with a sword, all other things being equal. My own fighting experience and observation tell me otherwise; however, I will readily admit that I do not have enough experience with weapons to know if this holds true for weapons (some or generally).
I can't speak for all weapons or styles for using them. However, for most swords, what 5e defines as Strength plays a large part in landing a blow even before you consider having to get past armour or layers of hide and muscle.
In actual combat, as opposed to what you might see in a lot of media or in weapon displays, a big, unopposed swing that strikes and continues is rare. The ability to control the weapon through leverage exerted is essential: the more force you can exert on the handle, the faster you can change direction of a swing and the less time and distance you need to generate a damaging attack.
So, outside of actual skill and technique, I would consider the attributes covered by Strength in 5e D&D as more important in landing a damaging hit.
Reflexes, balance and suchlike are also important in a fight, but if you had to simplify things down to D&D's level, using Str as the offensive stat, and Dex as the defensive stat makes more sense than the other way around.
In terms of whether I would change the rules to reflect this? Not really.
D&D exists to allow popular media tropes as well as realism. Apart from a few adjustments (removing finesse requirement for sneak attacks, allowing Str to be used with Bows etc) I personally see the value of finesse as a rules mechanic to allow certain character concepts despite viewing it as rather unrealistic.
If you did want to remove finesse, I would also suggest applying Dex to all armour, at least partially.
