The external power-on reset circuit is not needed for the AVR. However, be very cautious with reassigning the reset pin - it's also used to enable programming mode. There's a fuse in the AVR that can be set to allow use of RESET as a general-purpose I/O pin, but once that's done, the only way to reprogram the part is with high-voltage programming (which far fewer people have programmers for).
If you want to move to a bigger part, I'd suggest an ATtiny24. It only costs a few cents more than the '13, but has 14 pins rather than 8.