for 5e as a streamlined game, I agree you could remove the simple/martial distinction and it would have very little effect.
However, to the point as some have brought up, the intent of LevelUp is to increase choices/options. So with regards to this I support adding effects to weapons.
In my opinion this could be done one of two main ways.
First, additional effects can be added to weapons, such as maybe hammers that can knock prone on a crit, or axes that have a 19-20 crit range, etc. The difficulty of this is that it makes weapons as a whole more powerful. I'm not sure how this would be best balanced.
Second option, weapons gain additional maneuvers/attack types. This could either be added through class features or affixed to the weapon itself, gated by only those who have proficiency in the weapon gaining access to these maneuvers. To balance, these will need to have a downside to the alternate attacks. For example, an axe could have a "Slash" has a regular attack, and a "Hack", which has a 19-20 crit range, but also a -2 to attack. A longsword might have a "draw cut" as a regular attack or a "defensive strike" which has a -2 to attack, but grants +1 to AC vs melee attacks.