No, because then badly poisoned becomes overpowered. You'd need to balance it out by introducing a badly burned, which would also be very overpowering, plus you would need to introduce a move that simply causes badly burned status (to match Toxic).
Now we need a new move that just normal-poisons an enemy (like Will O Wisp), an item that normal-poisons the holder (like Flame Orb), and an ability to reduce the damage per turn (like Heatproof).
Plus, things like Toxic Boost Zangoose rise in usage because they can infect themselves with normal-poison without sacrificing extra health or physical attack. Poison and Steel type special attackers rise in usage because they are immune to the new special-dropping poison.
Finally, things like Pecha Berry and Antidote now cure this special-dropping poison too, so they will increase in value (not a huge metagame change, but still).
EDIT: Oh, and I almost forgot, now 1-layer Toxic Spikes cripples any special sweeper switching in!