Might be just me... but I don't see the hindrance of carrying a Simple 1d6 shortsword rather than a Martial 1d8 longsword to equate to as might as well being unarmed?
It's a point of damage less. How can that be such a grave issue? Haven't people been screaming about how the Flex weapon property sucks because it's only a point of damage gain? How can that one point of damage be a meaningless boon for weapon mastery but be this huge burden for proficiency?
Shortswords are simple. They are martial again. If short swords were simple, a lot of problems would be solved.
As it stands, bards, clerics, druids, monks, warlocks, sorcerers and wizards get simple only. (Druid and Cleric can upgrade to Martial via their order but let's assume we are looking primarily for a spellcaster with a backup weapon). If you're going to be a caster, you're prioritizing Dex over Str (often to the point Str is a dump stat). Lets take each class in order.
A Bard used to have the gold standard of finesse weapons: the rapier. Most bards have good dexterity scores because of skills and light armor, so the rapier was a no brainer. On the simple weapon chart, there is a single finesse weapon: the dagger. Every other simple weapon use Strength, and most cap at d6 1 handed, d8 two-handed. If a bard dumps Str (8) and has a moderate dex (14) his options are a 1d4+2 dagger (3 points of damage) or 1d8-1 two-handed spear (3 points of damage). There is almost NO reason to do either when you have firebolt on the arcane list, sacred flame on the divine, and shillelagh on the primal list, plus viscious mockery (upgraded to d6). The bard should leave his weapons at home and carry his lute two-handed.
The Cleric has a little more leeway since they get medium armor and shields, allowing them to splurge on Strength and carry a mace or spear 1h, both doing 1d6 damage. There is no reason though to pick the caster option over the martial though; 1 extra cantrip and a bonus on religion checks is =/= to martial weapons and heavy armor. And if you are picking the Protector, you might as well opt for a longsword or other d8 weapon.
The Druid already had little reason to ever carry a weapon; shillelagh and wild shape handled 99% of their combat needs. In the rare occasions they wanted a weapon, the scimitar was a finesse weapon which (like the bard) allowed them to use their moderate dex to dump Str (and dump Str on a druid is a no brainer choice since again, you're either casting or you're a bear). The Warden Choice is again superior to the Magician one, but that's only for the armor. If you are insisting on Magician, your options look a lot like the bard (or worse, since druids get shield proficiency which makes 2h a spear or staff less appealing. So again, if a 1d4 +2 dagger or 1d6-1 spear are your choices, just spam shillelagh.
The monk could warrant a whole novel on what weapons vs unarmed strike means. Rather than deal with the damage, I'll touch on the other issue: monk's can't qualify for any good feats. Going by the Experts UA feats, here are the feats they cannot use: Charger, Crossbow Expert, Dual Wielder, Fighting Style (any), Great Weapon Master, Mage Slayer, Mounted Combatant, Polearm Master, Sentinel, Sharpshooter. All of which they COULD have qualified for because they had a single martial weapon: the short sword. The monk is a fighting class with the combat options of a caster class.
Warlocks already have the best ranged attack in the game (eldritch blast) and unless you are going bladepact, there is no reason to ever carry a weapon. This was already a problem in 2014 and now it's spread to other classes.
Sorcerer and Wizard actually GAINED proficiency in additional weapons (greatclub, handaxe, light hammer, javelin mace, sickle, spear and shortbow). None are actually much better than the choices they already had. Again, dagger for finesse vs two-handed spear/staff is a wash since caster's dump strength and need dex for AC, and every ranged cantrip is superior to ranged weapons like light crossbows and shortbows.
As an honorable mention: the rogue gets simple weapons AND finesse martial weapons. This is because WotC realized that the only simple weapons you can sneak attack with are ranged weapons and daggers, so they tossed them a bone gave them rapier and shortsword. It's the only class that also breaks the simple or simple/martial by giving them simple + not full martial.
Ideally, this is how I would fix it:
1.) Move scimitar and shortsword to the simple list. They are d6 finesse weapons and aren't breaking anything on non-mastery characters.
2.) Give Bards the same base proficiency as Rogues: simple + Finesse martial.
3.) Allow a monk to treat unarmed strikes as martial weapons for purposes of picking feats.
Yes, I already left this feedback.