If you want a 16F628 with built-in A2D, have a look at the 16F819, basically a 628 with 5x10 bit A2D - and it has an 8MHz internal clock oscillator, although you can switch it to various different speeds - including 4MHz like the 628.
If you're only wanting an input pot, one output pin, and a start button (and perhaps a stop button), you could use the 12F675, an 8 pin FLASH PIC with 10 bit A2D. Same size as a 555, but will make a much better timer :lol:
As for the capacitor charging technique, it's basically how the joystick port on your PC works - there's no A2D on a joystick port, just the same capacitor charging technique!. It's certainly more than accurate enough for reading a pot.