It's different from, say, an AM or FM radio, which tunes the specific station in frequency. Instead a CDMA (or any other spread spectrum modulated signal) have a wide-band RF front-end because all the stations are occupying exactly the same width of spectrum. So all the stations are present in the RF and IF stages because no channel separation is performed until baseband: basically at the detector stage. In short: the radio does not separate any stations at the front-end at all.
Your intuition is correct: something "has to give" with this, and that is the noise figure of the RF section and thus the received SNR, apples-to-apples, isn't going to be as good as a radio that pre-selects a narrow channel in the RF stage.
But spread-spectrum is a wide band signal so you can't separate at RF anyway and all of the stations are occupying the same bandwidth/channel. Poorer SNR is the trade of going to digital modulation but especially spread-spectrum. As you add more channels the RF spectrum's noise floor rises and that does have an effect on the RF section.
The thing that is different though (for cellular) from AM/FM is the cell: the range between radios is small by design so the signal can be assured to be strong enough. Having a "problem" in this sense actually helps define the cell boundaries more strongly. This isn't an amateur radio "QRP" scenario. System assumes infrastructure of multiple cells and backchannel hand-off between cells always keeps RF strength high.
For other applications like WiFi, there is no expectation of wide coverage (kilometers) so signal strength is strong enough and is what it is at the fringes, only 10s of meters from the base station.
In phone line modems you have a similar situation but also a constrained physical bandwidth.
Think of it this way: you can't get something for nothing, and to get the higher digital bandwidth than spectrum width would normally allow have to be paid for, and you pay with SNR. This is part of the "http://en.wikipedia.org/wiki/Cliff_effect" " effect (combined with the threshold effect of digital data synchronization and error correction).