Correct microprocessor to replace Arduino Uno

The tiny85 doesn't sound like it has enough program memory.
It would fit in an atmega168, or probably an attiny1634, but there isn't much of a reason for the change. In quantity, you'd save a few cents per chip, maybe.

You can put a 328 on a board that only breaks out some of the pins, and it gets significantly smaller. Adafruit trinket pro comes to mind.