Depends on what you mean by digital receiver.
If you mean digital tuning (i.e. a radio with maybe an LCD display and push-buttons), that uses a PLL, so technically it is
still varying a C value, but by means of a varactor.
If you mean a software radio type architecture (currently not usually the type of radio you can buy at your local store)
again there are different ways. For example, a conventional front end and mixer may be used (i.e. still relying on a PLL or
maybe DDS or some combination), and then the IF may be sampled with a DAC, and then it's down to software to perform
the demodulation. Other types may directly take the RF input (not mixed to an IF) and software (DSP) filters are used to
select the channel.