I love the idea of those with martial weapon proficiency having special abilities linked to weapons as standard but the proliferation of saving throws would really drag down the game.
Simple, don't allow saving throws vs crit effects. The save was the target's AC, which the character critically overcame. You shouldn't get a save when the master swordsman beat your AC by 10, you just automatically bleed for their stat+proficiency bonus until you stop it.
Martial crits outside of specific situations (rogue/paladin smite) are pretty lame in 5E, so I see no real downside to throwing these on top of the paltry d8 your longsword fighter would normally add. If we really need to keep martials so simple a potato can play them, allow an extra d6 damage instead of the effect.
PF2E's crit effects are below for reference
Axe: Deal weapon damage to adjacent target if you would also hit their AC with your result
Bow: Pin target to adjacent surface. Target must spend bonus action and DC 10 athletics to move.
Club: Knock the target up to 10 feet back
Flail/Hammer: Target knocked prone
Knife: d6 persistent bleed damage
Polearm: Move target 5' in your choice
Shield: Knock target back 5'
Sling: Target loses one action next turn
Spear: Target is clumsy 1 until next turn (-1 AC and Dex based checks)
Sword: Target is flat footed until start of next turn
The nice thing about keeping critical success at 10+ AC is you avoid the situation where you can only be crit or missed, and it makes every "+1" count, as they increase the likelihood of critting/avoiding the crit.