What if it was "you can add this to any roll in place of your normal modifier, but you don't add your Proficiency Bonus"? This makes it the Luck stat.
Death save? Add luck. Attack roll when you're not good with weapons? Add luck. Trying to pick a lock when you don't know how? Add luck.
A wizard with a maxed luck (+5) makes an attack roll with a dagger... he only gets +5. The rogue (max dex & proficiency) gets +7 to +11, so the "lucky" character isn't ever better than a specialized character. Putting ASIs into Luck helps characters get lucky, but doesn't substitute for true proficiency or primary-stat ASIs.