Blue
Ravenous Bugblatter Beast of Traal
An arcane focus is a special item that costs a lot of money (it is 25× the cost of a quarterstaff). Do you really want to be whacking dirty monsters with it?
Depending on what it is made of, an arcane focus might be too fragile to use in combat (more than once, anyway). Combat might wreck its ornate decoration and arcane ingredients, making it useless for spellcasting.
Sounds like a question for the DM. I've had it come up at one table I played with and the DM was fine with it being sturdy but you could go the other way as well.
Personally I allow it as well - it both fits my vision of what Gandalf carries, but also mimics that magic item staffs can be used as quarterstaves (DMG pg 140 under Staffs). Plus you can have a Holy Symbol on a shield getting smashed by ogre's mauls without problem and it's the same cost, it doesn't seem like there's anything inherently weak as long as your base item is sufficient.
As for the psychological issue you mentioned: If the cost - 5gp - was too expensive to band on dirty monsters, we'd have to give up every martial melee weapon except the whip since all the others cost 5gp or more.