In the desert, when hundreds of thousands of people are all in close proximity, it's good for avoiding disease. Once in the promised land, it was a reminder of the holiness of God, but far more a ceremonial thing.
Jesus fulfils these regulations in the first 7 chapters of Mark by:
- Touching someone with a skin disease (1:40-45)
- Being touched by a bleeding woman (5:25-34)
- Touching a dead child (5:35-43)
- Explaining what real uncleanness is (7:14-23)
Real uncleanness is a matter of the heart - nothing going in can defile, only that which comes out. What goes in goes into the stomach (there's your eating animals allowed - explicit in the text (7:19b)) and passes through - out into the toilet. It's what comes out of the heart that makes people unclean - "evil thoughts, sexual immoralities, thefts, murders, adulteries, greed, evil actions, deceit, lewdness, stinginess, blasphemy, pride, and foolishness" (7:21-22).
Funnily enough, his 'what makes people unclean' come just before he goes and deals with Gentiles - the Syrophoenician woman, the deaf and dumb man (and others) and the 4000. Gentiles were considered unclean as well, and he's removed all the restriction.
Therefore we don't need to worry about falling foul of the clean/unclean rules because Jesus is willing and he can make us clean (Mk1:41). We don't need to worry about food laws, nor cleaning pots (more than is necessary to stop bacteria), or being Gentiles, as those things don't make us unclean. Rather the fulfilment of the laws about clean/unclean is found in Mk 7:20-23. That it's sin that makes us truly unclean - thankfully Jesus deals with that too - both in our justification and sanctification.
What's great is that Christians don't have to go through purification rituals to approach God, or even to be with his people. They've already been purified, washed clean in the blood of the Lamb.