Hardware issue. Servo outputs are created from PWM channels. Mega has more than Uno, hence can support more servos.
– Two 8-bit Timer/Counters with Separate Prescalers and Compare Modes
– One 16-bit Timer/Counter with Separate Prescaler, Compare Mode, and Capture Mode
– Real Time Counter with Separate Oscillator
– Six PWM Channels
vs
– Two 8-bit Timer/Counters with Separate Prescaler and Compare Mode
– Four 16-bit Timer/Counter with Separate Prescaler, Compare- and Capture Mode
– Real Time Counter with Separate Oscillator
– Four 8-bit PWM Channels
– Six/Twelve PWM Channels with Programmable Resolution from 2 to 16 Bits
(ATmega1281/2561, ATmega640/1280/2560)
I'd normally suggest an upgrade to an intermediate processor, Atmega1284P, but it has the same PWM complement as the '328P, so that doesn't help.
So - roll your own using blink without delay style programming to fake some more outputs, or stick with a Mega.