Isn't the whole point of Purell and the like to be able to wash your hands without needing a sink and traditional soap?
No.
Alcohol gels only sanitize. Sanitization means killing harmful bacteria. It does not remove dirt, grease, etc. More importantly, dirt/grease/biological material deacivates the sanitization effect, so the sanitizer is ineffective for unclean hands.
Where the hands are visibly contaminated after going to the bathroom, while the sanitizer may remove bacteria this is of limited value as:
In the case of urine contamination, urine doesn't contain bacteria/germs anyway - so the santizing effect is not necessary; second, the sanitizer won't do anything about any odor associated with it.
In the case of fecal contamination, if this is visible, you risk inactivating the sanitizer.
The only use for alcohol gels is for people who are at risk of getting trace contamination by infective material on a regular basis (e.g. healthcare workers). People whose hands get dirty should not use sanitizers.