The purpose of material components (which includes a spellcasting focus) is to control action economy.
To cast a spell you need a hand free to hold the components. To hit someone you need a hand free to hold the weapon. To defend yourself you need a hand free to hold the shield.
This means that there is a limit on how many different actions you can do; namely two (four, if you are a thri-keen).
If you allow one item to be used for two purposes then you are changing the action economy.
You also remove some of the consequences of choices. For example, if you end your turn with a sword and a shield in your hands then you can't cast a spell as a reaction. Your choice has a meaningful consequence.
A quarterstaff is 2sp. A staff focus (both the arcane one and the druidic one) is 5gp. This suggests that a focus is an ornate, crafted item, made from expensive materials (in other words, not just a big stick). I imagine that if you hit someone with your focus it will damage the item, possibly making it unsuitable to be a focus.
Inscribing a holy symbol on a shield costs 5gp, half the price of the original shield.
The 12 different types of focus are listed in the Adventuring Gear section, not the Weapons section. The focus is called a "staff" or a "wooden staff", not a "quarterstaff". I think this clearly indicates that focuses are not supposed to be weapons.