JacktheRabbit
Explorer
Personally I've always interpreted "proficient with martial weapons" to mean that you've trained with just some of them. Which ones? The ones you end up deciding to use, of course.
That said, I hope the Rapier dies. Not only does it not fit thematically, it is so clearly best-in-class for Dex users that it has become ubiquitous.
One solution is to max out 1H finesse weapons at 1d6 (and then offer some alternatives to the short sword).
The other solution...admittedly for 6e, not 5e...is to find a way for both Dex and Str to contribute to melee combat so that there's no such thing as "Finesse", and no easy stat dumping of one or the other. That is, it should be totally viable for a Dex fighter to use a Longsword, without simply replacing Str with Dex as the only combat stat. (I might bias the mechanics to slightly favor Str for Heavy weapons, and favor Dex for Light weapons, however.)
To me at least an easy solution would be the old edition rules that allowed you to add 1.5 strength bonus when wielding two handed weapons. This helps out make strength better for two handers but does not help the fact that a rapier by rules is generally better to build a character around than a long sword since Dex is straight out better than Str as a stat. So another balance could be that finesse cannot be used with a shield and finesse bonuses are maxed like AC bonus by the armor you wear. The logic being that finesse weapons require one to be able to move quickly and shields as well as heavy armor both impede ones ability to flit around with their rapier.