I guess my thought is that you are training with your next weapon [Edit: off camera] while progressing in levels, and every so often you achieve proficiency, just like spellcasters suddenly gain more spells at future levels.
Here's an idea:
For martial classes:
Starting at first level, you have learned to fight with single weapons. Choose either ammunition, bludgeoning, piercing, or slashing. You gain proficiency in non-ammunition weapons with that damage type or weapons with the ammunition property.
At 1st level, you have learned to use any such weapon that uses 1d4 or 1d6 damage dice. At 5th level, you learn to use 1d8 and 2d4 weapons; at 10th, 1d10 weapons; and at 15th, 1d12 and 2d6 weapons.
At 5th level, you learn to fight with two 1d4 weapons, and at 15th level, two 1d6 weapons.
For full spellcaster classes:
Starting at first level, you have learned to fight with single weapons. Choose either ammunition, bludgeoning, piercing, or slashing. You gain proficiency in non-ammunition weapons with that damage type or weapons with the ammunition property.
At 1st level, you have learned to use any such weapon that uses 1d4 damage dice. At 8th level, you learn to use 1d6 weapons; and at 16th, 1d8 and 2d4 weapons. At 16th level you also learn to fight with two 1d4 weapons.
Edit: I haven't played any version other than 5e. I didn't know it used to be different. Do martial classes not like having that advancement and having weapon choice matter?