It is possible to go too far in the other direction:
In some cases perhaps it might be seen as difficult or even impossible to find proof of someone's illegal status. In that case their right to stay would be recognised. If in some cases people would be granted that status when they were not entitled, that's how it goes. Better by far to err in that direction than the other way, as has been done till now.
If an illegal immigrant enters the UK tonight on the back of a lorry, and disappears into a local neighbourhood without being captured on CCTV. When he reappears tomorrow morning, what 'evidence can be found to show that he is an illegal immigrant? Are we to take his word for his legal status?

Lack of proof that he is here illegally is not proof that he is here legally.
Similarly:
Lack of proof that he is here legally is not proof that he is here illegally.

Each case needs to be treated on its own merits.